Question 23350: Normalize the Candidate before Checking Nonzero Square Roots

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0045-23350 · q-20261014-advance-20261111-r06-quadratic-residue-0050

Original question and givens

Use bounded odd prime p=101 and the declared nonzero quadratic-residue convention. Enumerate every x=1..p−1 and x^2 modulo p; normalize input classes into0..p−1. Zero is excluded from nonzero residues. Do not assume an Euler-criterion or Legendre-symbol result. Original a=25. Give (p, a, normalized_class, nonzero_roots, nonzero_residue, certificate).

G001 — Use bounded odd prime p=101 and the declared nonzero quadratic-residue convention. Enumerate every x=1..p−1 and x^2 modulo p; normalize input classes into0..p−1. Zero is excluded from nonzero residues. Do not assume an Euler-criterion or Legendre-symbol result. Original a=25. Give (p, a, normalized_class, nonzero_roots, nonzero_residue, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

mko-quadratic-residue

Step-by-step derivation

S001 · implies

Before

source operation=quadratic-membership-witness;original snapshot and mathematics retained

After

explicit ordered source output fields=[p, a, normalized_class, nonzero_roots, nonzero_residue, certificate]

Conditions: none

Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

Quadratic ORIGINAL modulus:ORIGINAL integer=101;complete possible trial divisors=[2, 3, 4, 5, 6, 7, 8, 9, 10]

After

ALL [divisor,remainder] rows=[[2, 1], [3, 2], [4, 1], [5, 1], [6, 5], [7, 3], [8, 5], [9, 2], [10, 1]];prime=Yes;a composite>1 has a factor at most its square root

Conditions: none

Check the entire square-root divisor range.The empty range qualifies prime2,but not0 or1.Samples,square tables or an Euler criterion do not replace primality qualification.

Step identifiers and dependencies

Rule: finite_prime_ALL_trial_divisors_through_square_root · Uses: G001, S001

S003 · calculation

Before

Quadratic ORIGINAL nonzero root1 square := mul(1, 1)

After

Quadratic ORIGINAL nonzero root1 square = 1

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

ORIGINAL nonzero root x=1 in1..100;actual square=1

After

Euclidean square=0*101+1;nonzero square class=1;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S003

S005 · calculation

Before

Quadratic ORIGINAL nonzero root2 square := mul(2, 2)

After

Quadratic ORIGINAL nonzero root2 square = 4

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

ORIGINAL nonzero root x=2 in1..100;actual square=4

After

Euclidean square=0*101+4;nonzero square class=4;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S005

S007 · calculation

Before

Quadratic ORIGINAL nonzero root3 square := mul(3, 3)

After

Quadratic ORIGINAL nonzero root3 square = 9

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

ORIGINAL nonzero root x=3 in1..100;actual square=9

After

Euclidean square=0*101+9;nonzero square class=9;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S007

S009 · calculation

Before

Quadratic ORIGINAL nonzero root4 square := mul(4, 4)

After

Quadratic ORIGINAL nonzero root4 square = 16

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

ORIGINAL nonzero root x=4 in1..100;actual square=16

After

Euclidean square=0*101+16;nonzero square class=16;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S009

S011 · calculation

Before

Quadratic ORIGINAL nonzero root5 square := mul(5, 5)

After

Quadratic ORIGINAL nonzero root5 square = 25

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

ORIGINAL nonzero root x=5 in1..100;actual square=25

After

Euclidean square=0*101+25;nonzero square class=25;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S011

S013 · calculation

Before

Quadratic ORIGINAL nonzero root6 square := mul(6, 6)

After

Quadratic ORIGINAL nonzero root6 square = 36

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

ORIGINAL nonzero root x=6 in1..100;actual square=36

After

Euclidean square=0*101+36;nonzero square class=36;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S013

S015 · calculation

Before

Quadratic ORIGINAL nonzero root7 square := mul(7, 7)

After

Quadratic ORIGINAL nonzero root7 square = 49

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

ORIGINAL nonzero root x=7 in1..100;actual square=49

After

Euclidean square=0*101+49;nonzero square class=49;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S015

S017 · calculation

Before

Quadratic ORIGINAL nonzero root8 square := mul(8, 8)

After

Quadratic ORIGINAL nonzero root8 square = 64

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

ORIGINAL nonzero root x=8 in1..100;actual square=64

After

Euclidean square=0*101+64;nonzero square class=64;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S017

S019 · calculation

Before

Quadratic ORIGINAL nonzero root9 square := mul(9, 9)

After

Quadratic ORIGINAL nonzero root9 square = 81

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

ORIGINAL nonzero root x=9 in1..100;actual square=81

After

Euclidean square=0*101+81;nonzero square class=81;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S019

S021 · calculation

Before

Quadratic ORIGINAL nonzero root10 square := mul(10, 10)

After

Quadratic ORIGINAL nonzero root10 square = 100

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

ORIGINAL nonzero root x=10 in1..100;actual square=100

After

Euclidean square=0*101+100;nonzero square class=100;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S021

S023 · calculation

Before

Quadratic ORIGINAL nonzero root11 square := mul(11, 11)

After

Quadratic ORIGINAL nonzero root11 square = 121

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

ORIGINAL nonzero root x=11 in1..100;actual square=121

After

Euclidean square=1*101+20;nonzero square class=20;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S023

S025 · calculation

Before

Quadratic ORIGINAL nonzero root12 square := mul(12, 12)

After

Quadratic ORIGINAL nonzero root12 square = 144

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

ORIGINAL nonzero root x=12 in1..100;actual square=144

After

Euclidean square=1*101+43;nonzero square class=43;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S025

S027 · calculation

Before

Quadratic ORIGINAL nonzero root13 square := mul(13, 13)

After

Quadratic ORIGINAL nonzero root13 square = 169

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

ORIGINAL nonzero root x=13 in1..100;actual square=169

After

Euclidean square=1*101+68;nonzero square class=68;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S027

S029 · calculation

Before

Quadratic ORIGINAL nonzero root14 square := mul(14, 14)

After

Quadratic ORIGINAL nonzero root14 square = 196

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

ORIGINAL nonzero root x=14 in1..100;actual square=196

After

Euclidean square=1*101+95;nonzero square class=95;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S029

S031 · calculation

Before

Quadratic ORIGINAL nonzero root15 square := mul(15, 15)

After

Quadratic ORIGINAL nonzero root15 square = 225

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · implies

Before

ORIGINAL nonzero root x=15 in1..100;actual square=225

After

Euclidean square=2*101+23;nonzero square class=23;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S031

S033 · calculation

Before

Quadratic ORIGINAL nonzero root16 square := mul(16, 16)

After

Quadratic ORIGINAL nonzero root16 square = 256

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · implies

Before

ORIGINAL nonzero root x=16 in1..100;actual square=256

After

Euclidean square=2*101+54;nonzero square class=54;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S033

S035 · calculation

Before

Quadratic ORIGINAL nonzero root17 square := mul(17, 17)

After

Quadratic ORIGINAL nonzero root17 square = 289

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · implies

Before

ORIGINAL nonzero root x=17 in1..100;actual square=289

After

Euclidean square=2*101+87;nonzero square class=87;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S035

S037 · calculation

Before

Quadratic ORIGINAL nonzero root18 square := mul(18, 18)

After

Quadratic ORIGINAL nonzero root18 square = 324

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

ORIGINAL nonzero root x=18 in1..100;actual square=324

After

Euclidean square=3*101+21;nonzero square class=21;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S037

S039 · calculation

Before

Quadratic ORIGINAL nonzero root19 square := mul(19, 19)

After

Quadratic ORIGINAL nonzero root19 square = 361

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · implies

Before

ORIGINAL nonzero root x=19 in1..100;actual square=361

After

Euclidean square=3*101+58;nonzero square class=58;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S039

S041 · calculation

Before

Quadratic ORIGINAL nonzero root20 square := mul(20, 20)

After

Quadratic ORIGINAL nonzero root20 square = 400

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · implies

Before

ORIGINAL nonzero root x=20 in1..100;actual square=400

After

Euclidean square=3*101+97;nonzero square class=97;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S041

S043 · calculation

Before

Quadratic ORIGINAL nonzero root21 square := mul(21, 21)

After

Quadratic ORIGINAL nonzero root21 square = 441

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · implies

Before

ORIGINAL nonzero root x=21 in1..100;actual square=441

After

Euclidean square=4*101+37;nonzero square class=37;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S043

S045 · calculation

Before

Quadratic ORIGINAL nonzero root22 square := mul(22, 22)

After

Quadratic ORIGINAL nonzero root22 square = 484

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · implies

Before

ORIGINAL nonzero root x=22 in1..100;actual square=484

After

Euclidean square=4*101+80;nonzero square class=80;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S045

S047 · calculation

Before

Quadratic ORIGINAL nonzero root23 square := mul(23, 23)

After

Quadratic ORIGINAL nonzero root23 square = 529

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · implies

Before

ORIGINAL nonzero root x=23 in1..100;actual square=529

After

Euclidean square=5*101+24;nonzero square class=24;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S047

S049 · calculation

Before

Quadratic ORIGINAL nonzero root24 square := mul(24, 24)

After

Quadratic ORIGINAL nonzero root24 square = 576

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · implies

Before

ORIGINAL nonzero root x=24 in1..100;actual square=576

After

Euclidean square=5*101+71;nonzero square class=71;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S049

S051 · calculation

Before

Quadratic ORIGINAL nonzero root25 square := mul(25, 25)

After

Quadratic ORIGINAL nonzero root25 square = 625

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · implies

Before

ORIGINAL nonzero root x=25 in1..100;actual square=625

After

Euclidean square=6*101+19;nonzero square class=19;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S051

S053 · calculation

Before

Quadratic ORIGINAL nonzero root26 square := mul(26, 26)

After

Quadratic ORIGINAL nonzero root26 square = 676

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · implies

Before

ORIGINAL nonzero root x=26 in1..100;actual square=676

After

Euclidean square=6*101+70;nonzero square class=70;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S053

S055 · calculation

Before

Quadratic ORIGINAL nonzero root27 square := mul(27, 27)

After

Quadratic ORIGINAL nonzero root27 square = 729

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · implies

Before

ORIGINAL nonzero root x=27 in1..100;actual square=729

After

Euclidean square=7*101+22;nonzero square class=22;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S055

S057 · calculation

Before

Quadratic ORIGINAL nonzero root28 square := mul(28, 28)

After

Quadratic ORIGINAL nonzero root28 square = 784

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · implies

Before

ORIGINAL nonzero root x=28 in1..100;actual square=784

After

Euclidean square=7*101+77;nonzero square class=77;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S057

S059 · calculation

Before

Quadratic ORIGINAL nonzero root29 square := mul(29, 29)

After

Quadratic ORIGINAL nonzero root29 square = 841

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · implies

Before

ORIGINAL nonzero root x=29 in1..100;actual square=841

After

Euclidean square=8*101+33;nonzero square class=33;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S059

S061 · calculation

Before

Quadratic ORIGINAL nonzero root30 square := mul(30, 30)

After

Quadratic ORIGINAL nonzero root30 square = 900

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · implies

Before

ORIGINAL nonzero root x=30 in1..100;actual square=900

After

Euclidean square=8*101+92;nonzero square class=92;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S061

S063 · calculation

Before

Quadratic ORIGINAL nonzero root31 square := mul(31, 31)

After

Quadratic ORIGINAL nonzero root31 square = 961

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · implies

Before

ORIGINAL nonzero root x=31 in1..100;actual square=961

After

Euclidean square=9*101+52;nonzero square class=52;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S063

S065 · calculation

Before

Quadratic ORIGINAL nonzero root32 square := mul(32, 32)

After

Quadratic ORIGINAL nonzero root32 square = 1024

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · implies

Before

ORIGINAL nonzero root x=32 in1..100;actual square=1024

After

Euclidean square=10*101+14;nonzero square class=14;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S065

S067 · calculation

Before

Quadratic ORIGINAL nonzero root33 square := mul(33, 33)

After

Quadratic ORIGINAL nonzero root33 square = 1089

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · implies

Before

ORIGINAL nonzero root x=33 in1..100;actual square=1089

After

Euclidean square=10*101+79;nonzero square class=79;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S067

S069 · calculation

Before

Quadratic ORIGINAL nonzero root34 square := mul(34, 34)

After

Quadratic ORIGINAL nonzero root34 square = 1156

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · implies

Before

ORIGINAL nonzero root x=34 in1..100;actual square=1156

After

Euclidean square=11*101+45;nonzero square class=45;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S069

S071 · calculation

Before

Quadratic ORIGINAL nonzero root35 square := mul(35, 35)

After

Quadratic ORIGINAL nonzero root35 square = 1225

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · implies

Before

ORIGINAL nonzero root x=35 in1..100;actual square=1225

After

Euclidean square=12*101+13;nonzero square class=13;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S071

S073 · calculation

Before

Quadratic ORIGINAL nonzero root36 square := mul(36, 36)

After

Quadratic ORIGINAL nonzero root36 square = 1296

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · implies

Before

ORIGINAL nonzero root x=36 in1..100;actual square=1296

After

Euclidean square=12*101+84;nonzero square class=84;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S073

S075 · calculation

Before

Quadratic ORIGINAL nonzero root37 square := mul(37, 37)

After

Quadratic ORIGINAL nonzero root37 square = 1369

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · implies

Before

ORIGINAL nonzero root x=37 in1..100;actual square=1369

After

Euclidean square=13*101+56;nonzero square class=56;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S075

S077 · calculation

Before

Quadratic ORIGINAL nonzero root38 square := mul(38, 38)

After

Quadratic ORIGINAL nonzero root38 square = 1444

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S076

S078 · implies

Before

ORIGINAL nonzero root x=38 in1..100;actual square=1444

After

Euclidean square=14*101+30;nonzero square class=30;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S077

S079 · calculation

Before

Quadratic ORIGINAL nonzero root39 square := mul(39, 39)

After

Quadratic ORIGINAL nonzero root39 square = 1521

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S078

S080 · implies

Before

ORIGINAL nonzero root x=39 in1..100;actual square=1521

After

Euclidean square=15*101+6;nonzero square class=6;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S079

S081 · calculation

Before

Quadratic ORIGINAL nonzero root40 square := mul(40, 40)

After

Quadratic ORIGINAL nonzero root40 square = 1600

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S080

S082 · implies

Before

ORIGINAL nonzero root x=40 in1..100;actual square=1600

After

Euclidean square=15*101+85;nonzero square class=85;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S081

S083 · calculation

Before

Quadratic ORIGINAL nonzero root41 square := mul(41, 41)

After

Quadratic ORIGINAL nonzero root41 square = 1681

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S082

S084 · implies

Before

ORIGINAL nonzero root x=41 in1..100;actual square=1681

After

Euclidean square=16*101+65;nonzero square class=65;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S083

S085 · calculation

Before

Quadratic ORIGINAL nonzero root42 square := mul(42, 42)

After

Quadratic ORIGINAL nonzero root42 square = 1764

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S084

S086 · implies

Before

ORIGINAL nonzero root x=42 in1..100;actual square=1764

After

Euclidean square=17*101+47;nonzero square class=47;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S085

S087 · calculation

Before

Quadratic ORIGINAL nonzero root43 square := mul(43, 43)

After

Quadratic ORIGINAL nonzero root43 square = 1849

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S086

S088 · implies

Before

ORIGINAL nonzero root x=43 in1..100;actual square=1849

After

Euclidean square=18*101+31;nonzero square class=31;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S087

S089 · calculation

Before

Quadratic ORIGINAL nonzero root44 square := mul(44, 44)

After

Quadratic ORIGINAL nonzero root44 square = 1936

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S088

S090 · implies

Before

ORIGINAL nonzero root x=44 in1..100;actual square=1936

After

Euclidean square=19*101+17;nonzero square class=17;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S089

S091 · calculation

Before

Quadratic ORIGINAL nonzero root45 square := mul(45, 45)

After

Quadratic ORIGINAL nonzero root45 square = 2025

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S090

S092 · implies

Before

ORIGINAL nonzero root x=45 in1..100;actual square=2025

After

Euclidean square=20*101+5;nonzero square class=5;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S091

S093 · calculation

Before

Quadratic ORIGINAL nonzero root46 square := mul(46, 46)

After

Quadratic ORIGINAL nonzero root46 square = 2116

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S092

S094 · implies

Before

ORIGINAL nonzero root x=46 in1..100;actual square=2116

After

Euclidean square=20*101+96;nonzero square class=96;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S093

S095 · calculation

Before

Quadratic ORIGINAL nonzero root47 square := mul(47, 47)

After

Quadratic ORIGINAL nonzero root47 square = 2209

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S094

S096 · implies

Before

ORIGINAL nonzero root x=47 in1..100;actual square=2209

After

Euclidean square=21*101+88;nonzero square class=88;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S095

S097 · calculation

Before

Quadratic ORIGINAL nonzero root48 square := mul(48, 48)

After

Quadratic ORIGINAL nonzero root48 square = 2304

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S096

S098 · implies

Before

ORIGINAL nonzero root x=48 in1..100;actual square=2304

After

Euclidean square=22*101+82;nonzero square class=82;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S097

S099 · calculation

Before

Quadratic ORIGINAL nonzero root49 square := mul(49, 49)

After

Quadratic ORIGINAL nonzero root49 square = 2401

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S098

S100 · implies

Before

ORIGINAL nonzero root x=49 in1..100;actual square=2401

After

Euclidean square=23*101+78;nonzero square class=78;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S099

S101 · calculation

Before

Quadratic ORIGINAL nonzero root50 square := mul(50, 50)

After

Quadratic ORIGINAL nonzero root50 square = 2500

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S100

S102 · implies

Before

ORIGINAL nonzero root x=50 in1..100;actual square=2500

After

Euclidean square=24*101+76;nonzero square class=76;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S101

S103 · calculation

Before

Quadratic ORIGINAL nonzero root51 square := mul(51, 51)

After

Quadratic ORIGINAL nonzero root51 square = 2601

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S102

S104 · implies

Before

ORIGINAL nonzero root x=51 in1..100;actual square=2601

After

Euclidean square=25*101+76;nonzero square class=76;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S103

S105 · calculation

Before

Quadratic ORIGINAL nonzero root52 square := mul(52, 52)

After

Quadratic ORIGINAL nonzero root52 square = 2704

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S104

S106 · implies

Before

ORIGINAL nonzero root x=52 in1..100;actual square=2704

After

Euclidean square=26*101+78;nonzero square class=78;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S105

S107 · calculation

Before

Quadratic ORIGINAL nonzero root53 square := mul(53, 53)

After

Quadratic ORIGINAL nonzero root53 square = 2809

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S106

S108 · implies

Before

ORIGINAL nonzero root x=53 in1..100;actual square=2809

After

Euclidean square=27*101+82;nonzero square class=82;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S107

S109 · calculation

Before

Quadratic ORIGINAL nonzero root54 square := mul(54, 54)

After

Quadratic ORIGINAL nonzero root54 square = 2916

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S108

S110 · implies

Before

ORIGINAL nonzero root x=54 in1..100;actual square=2916

After

Euclidean square=28*101+88;nonzero square class=88;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S109

S111 · calculation

Before

Quadratic ORIGINAL nonzero root55 square := mul(55, 55)

After

Quadratic ORIGINAL nonzero root55 square = 3025

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S110

S112 · implies

Before

ORIGINAL nonzero root x=55 in1..100;actual square=3025

After

Euclidean square=29*101+96;nonzero square class=96;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S111

S113 · calculation

Before

Quadratic ORIGINAL nonzero root56 square := mul(56, 56)

After

Quadratic ORIGINAL nonzero root56 square = 3136

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S112

S114 · implies

Before

ORIGINAL nonzero root x=56 in1..100;actual square=3136

After

Euclidean square=31*101+5;nonzero square class=5;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S113

S115 · calculation

Before

Quadratic ORIGINAL nonzero root57 square := mul(57, 57)

After

Quadratic ORIGINAL nonzero root57 square = 3249

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S114

S116 · implies

Before

ORIGINAL nonzero root x=57 in1..100;actual square=3249

After

Euclidean square=32*101+17;nonzero square class=17;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S115

S117 · calculation

Before

Quadratic ORIGINAL nonzero root58 square := mul(58, 58)

After

Quadratic ORIGINAL nonzero root58 square = 3364

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S116

S118 · implies

Before

ORIGINAL nonzero root x=58 in1..100;actual square=3364

After

Euclidean square=33*101+31;nonzero square class=31;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S117

S119 · calculation

Before

Quadratic ORIGINAL nonzero root59 square := mul(59, 59)

After

Quadratic ORIGINAL nonzero root59 square = 3481

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S118

S120 · implies

Before

ORIGINAL nonzero root x=59 in1..100;actual square=3481

After

Euclidean square=34*101+47;nonzero square class=47;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S119

S121 · calculation

Before

Quadratic ORIGINAL nonzero root60 square := mul(60, 60)

After

Quadratic ORIGINAL nonzero root60 square = 3600

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S120

S122 · implies

Before

ORIGINAL nonzero root x=60 in1..100;actual square=3600

After

Euclidean square=35*101+65;nonzero square class=65;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S121

S123 · calculation

Before

Quadratic ORIGINAL nonzero root61 square := mul(61, 61)

After

Quadratic ORIGINAL nonzero root61 square = 3721

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S122

S124 · implies

Before

ORIGINAL nonzero root x=61 in1..100;actual square=3721

After

Euclidean square=36*101+85;nonzero square class=85;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S123

S125 · calculation

Before

Quadratic ORIGINAL nonzero root62 square := mul(62, 62)

After

Quadratic ORIGINAL nonzero root62 square = 3844

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S124

S126 · implies

Before

ORIGINAL nonzero root x=62 in1..100;actual square=3844

After

Euclidean square=38*101+6;nonzero square class=6;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S125

S127 · calculation

Before

Quadratic ORIGINAL nonzero root63 square := mul(63, 63)

After

Quadratic ORIGINAL nonzero root63 square = 3969

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S126

S128 · implies

Before

ORIGINAL nonzero root x=63 in1..100;actual square=3969

After

Euclidean square=39*101+30;nonzero square class=30;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S127

S129 · calculation

Before

Quadratic ORIGINAL nonzero root64 square := mul(64, 64)

After

Quadratic ORIGINAL nonzero root64 square = 4096

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S128

S130 · implies

Before

ORIGINAL nonzero root x=64 in1..100;actual square=4096

After

Euclidean square=40*101+56;nonzero square class=56;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S129

S131 · calculation

Before

Quadratic ORIGINAL nonzero root65 square := mul(65, 65)

After

Quadratic ORIGINAL nonzero root65 square = 4225

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S130

S132 · implies

Before

ORIGINAL nonzero root x=65 in1..100;actual square=4225

After

Euclidean square=41*101+84;nonzero square class=84;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S131

S133 · calculation

Before

Quadratic ORIGINAL nonzero root66 square := mul(66, 66)

After

Quadratic ORIGINAL nonzero root66 square = 4356

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S132

S134 · implies

Before

ORIGINAL nonzero root x=66 in1..100;actual square=4356

After

Euclidean square=43*101+13;nonzero square class=13;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S133

S135 · calculation

Before

Quadratic ORIGINAL nonzero root67 square := mul(67, 67)

After

Quadratic ORIGINAL nonzero root67 square = 4489

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S134

S136 · implies

Before

ORIGINAL nonzero root x=67 in1..100;actual square=4489

After

Euclidean square=44*101+45;nonzero square class=45;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S135

S137 · calculation

Before

Quadratic ORIGINAL nonzero root68 square := mul(68, 68)

After

Quadratic ORIGINAL nonzero root68 square = 4624

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S136

S138 · implies

Before

ORIGINAL nonzero root x=68 in1..100;actual square=4624

After

Euclidean square=45*101+79;nonzero square class=79;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S137

S139 · calculation

Before

Quadratic ORIGINAL nonzero root69 square := mul(69, 69)

After

Quadratic ORIGINAL nonzero root69 square = 4761

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S138

S140 · implies

Before

ORIGINAL nonzero root x=69 in1..100;actual square=4761

After

Euclidean square=47*101+14;nonzero square class=14;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S139

S141 · calculation

Before

Quadratic ORIGINAL nonzero root70 square := mul(70, 70)

After

Quadratic ORIGINAL nonzero root70 square = 4900

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S140

S142 · implies

Before

ORIGINAL nonzero root x=70 in1..100;actual square=4900

After

Euclidean square=48*101+52;nonzero square class=52;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S141

S143 · calculation

Before

Quadratic ORIGINAL nonzero root71 square := mul(71, 71)

After

Quadratic ORIGINAL nonzero root71 square = 5041

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S142

S144 · implies

Before

ORIGINAL nonzero root x=71 in1..100;actual square=5041

After

Euclidean square=49*101+92;nonzero square class=92;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S143

S145 · calculation

Before

Quadratic ORIGINAL nonzero root72 square := mul(72, 72)

After

Quadratic ORIGINAL nonzero root72 square = 5184

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S144

S146 · implies

Before

ORIGINAL nonzero root x=72 in1..100;actual square=5184

After

Euclidean square=51*101+33;nonzero square class=33;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S145

S147 · calculation

Before

Quadratic ORIGINAL nonzero root73 square := mul(73, 73)

After

Quadratic ORIGINAL nonzero root73 square = 5329

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S146

S148 · implies

Before

ORIGINAL nonzero root x=73 in1..100;actual square=5329

After

Euclidean square=52*101+77;nonzero square class=77;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S147

S149 · calculation

Before

Quadratic ORIGINAL nonzero root74 square := mul(74, 74)

After

Quadratic ORIGINAL nonzero root74 square = 5476

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S148

S150 · implies

Before

ORIGINAL nonzero root x=74 in1..100;actual square=5476

After

Euclidean square=54*101+22;nonzero square class=22;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S149

S151 · calculation

Before

Quadratic ORIGINAL nonzero root75 square := mul(75, 75)

After

Quadratic ORIGINAL nonzero root75 square = 5625

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S150

S152 · implies

Before

ORIGINAL nonzero root x=75 in1..100;actual square=5625

After

Euclidean square=55*101+70;nonzero square class=70;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S151

S153 · calculation

Before

Quadratic ORIGINAL nonzero root76 square := mul(76, 76)

After

Quadratic ORIGINAL nonzero root76 square = 5776

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S152

S154 · implies

Before

ORIGINAL nonzero root x=76 in1..100;actual square=5776

After

Euclidean square=57*101+19;nonzero square class=19;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S153

S155 · calculation

Before

Quadratic ORIGINAL nonzero root77 square := mul(77, 77)

After

Quadratic ORIGINAL nonzero root77 square = 5929

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S154

S156 · implies

Before

ORIGINAL nonzero root x=77 in1..100;actual square=5929

After

Euclidean square=58*101+71;nonzero square class=71;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S155

S157 · calculation

Before

Quadratic ORIGINAL nonzero root78 square := mul(78, 78)

After

Quadratic ORIGINAL nonzero root78 square = 6084

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S156

S158 · implies

Before

ORIGINAL nonzero root x=78 in1..100;actual square=6084

After

Euclidean square=60*101+24;nonzero square class=24;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S157

S159 · calculation

Before

Quadratic ORIGINAL nonzero root79 square := mul(79, 79)

After

Quadratic ORIGINAL nonzero root79 square = 6241

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S158

S160 · implies

Before

ORIGINAL nonzero root x=79 in1..100;actual square=6241

After

Euclidean square=61*101+80;nonzero square class=80;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S159

S161 · calculation

Before

Quadratic ORIGINAL nonzero root80 square := mul(80, 80)

After

Quadratic ORIGINAL nonzero root80 square = 6400

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S160

S162 · implies

Before

ORIGINAL nonzero root x=80 in1..100;actual square=6400

After

Euclidean square=63*101+37;nonzero square class=37;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S161

S163 · calculation

Before

Quadratic ORIGINAL nonzero root81 square := mul(81, 81)

After

Quadratic ORIGINAL nonzero root81 square = 6561

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S162

S164 · implies

Before

ORIGINAL nonzero root x=81 in1..100;actual square=6561

After

Euclidean square=64*101+97;nonzero square class=97;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S163

S165 · calculation

Before

Quadratic ORIGINAL nonzero root82 square := mul(82, 82)

After

Quadratic ORIGINAL nonzero root82 square = 6724

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S164

S166 · implies

Before

ORIGINAL nonzero root x=82 in1..100;actual square=6724

After

Euclidean square=66*101+58;nonzero square class=58;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S165

S167 · calculation

Before

Quadratic ORIGINAL nonzero root83 square := mul(83, 83)

After

Quadratic ORIGINAL nonzero root83 square = 6889

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S166

S168 · implies

Before

ORIGINAL nonzero root x=83 in1..100;actual square=6889

After

Euclidean square=68*101+21;nonzero square class=21;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S167

S169 · calculation

Before

Quadratic ORIGINAL nonzero root84 square := mul(84, 84)

After

Quadratic ORIGINAL nonzero root84 square = 7056

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S168

S170 · implies

Before

ORIGINAL nonzero root x=84 in1..100;actual square=7056

After

Euclidean square=69*101+87;nonzero square class=87;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S169

S171 · calculation

Before

Quadratic ORIGINAL nonzero root85 square := mul(85, 85)

After

Quadratic ORIGINAL nonzero root85 square = 7225

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S170

S172 · implies

Before

ORIGINAL nonzero root x=85 in1..100;actual square=7225

After

Euclidean square=71*101+54;nonzero square class=54;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S171

S173 · calculation

Before

Quadratic ORIGINAL nonzero root86 square := mul(86, 86)

After

Quadratic ORIGINAL nonzero root86 square = 7396

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S172

S174 · implies

Before

ORIGINAL nonzero root x=86 in1..100;actual square=7396

After

Euclidean square=73*101+23;nonzero square class=23;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S173

S175 · calculation

Before

Quadratic ORIGINAL nonzero root87 square := mul(87, 87)

After

Quadratic ORIGINAL nonzero root87 square = 7569

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S174

S176 · implies

Before

ORIGINAL nonzero root x=87 in1..100;actual square=7569

After

Euclidean square=74*101+95;nonzero square class=95;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S175

S177 · calculation

Before

Quadratic ORIGINAL nonzero root88 square := mul(88, 88)

After

Quadratic ORIGINAL nonzero root88 square = 7744

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S176

S178 · implies

Before

ORIGINAL nonzero root x=88 in1..100;actual square=7744

After

Euclidean square=76*101+68;nonzero square class=68;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S177

S179 · calculation

Before

Quadratic ORIGINAL nonzero root89 square := mul(89, 89)

After

Quadratic ORIGINAL nonzero root89 square = 7921

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S178

S180 · implies

Before

ORIGINAL nonzero root x=89 in1..100;actual square=7921

After

Euclidean square=78*101+43;nonzero square class=43;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S179

S181 · calculation

Before

Quadratic ORIGINAL nonzero root90 square := mul(90, 90)

After

Quadratic ORIGINAL nonzero root90 square = 8100

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S180

S182 · implies

Before

ORIGINAL nonzero root x=90 in1..100;actual square=8100

After

Euclidean square=80*101+20;nonzero square class=20;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S181

S183 · calculation

Before

Quadratic ORIGINAL nonzero root91 square := mul(91, 91)

After

Quadratic ORIGINAL nonzero root91 square = 8281

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S182

S184 · implies

Before

ORIGINAL nonzero root x=91 in1..100;actual square=8281

After

Euclidean square=81*101+100;nonzero square class=100;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S183

S185 · calculation

Before

Quadratic ORIGINAL nonzero root92 square := mul(92, 92)

After

Quadratic ORIGINAL nonzero root92 square = 8464

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S184

S186 · implies

Before

ORIGINAL nonzero root x=92 in1..100;actual square=8464

After

Euclidean square=83*101+81;nonzero square class=81;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S185

S187 · calculation

Before

Quadratic ORIGINAL nonzero root93 square := mul(93, 93)

After

Quadratic ORIGINAL nonzero root93 square = 8649

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S186

S188 · implies

Before

ORIGINAL nonzero root x=93 in1..100;actual square=8649

After

Euclidean square=85*101+64;nonzero square class=64;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S187

S189 · calculation

Before

Quadratic ORIGINAL nonzero root94 square := mul(94, 94)

After

Quadratic ORIGINAL nonzero root94 square = 8836

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S188

S190 · implies

Before

ORIGINAL nonzero root x=94 in1..100;actual square=8836

After

Euclidean square=87*101+49;nonzero square class=49;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S189

S191 · calculation

Before

Quadratic ORIGINAL nonzero root95 square := mul(95, 95)

After

Quadratic ORIGINAL nonzero root95 square = 9025

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S190

S192 · implies

Before

ORIGINAL nonzero root x=95 in1..100;actual square=9025

After

Euclidean square=89*101+36;nonzero square class=36;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S191

S193 · calculation

Before

Quadratic ORIGINAL nonzero root96 square := mul(96, 96)

After

Quadratic ORIGINAL nonzero root96 square = 9216

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S192

S194 · implies

Before

ORIGINAL nonzero root x=96 in1..100;actual square=9216

After

Euclidean square=91*101+25;nonzero square class=25;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S193

S195 · calculation

Before

Quadratic ORIGINAL nonzero root97 square := mul(97, 97)

After

Quadratic ORIGINAL nonzero root97 square = 9409

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S194

S196 · implies

Before

ORIGINAL nonzero root x=97 in1..100;actual square=9409

After

Euclidean square=93*101+16;nonzero square class=16;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S195

S197 · calculation

Before

Quadratic ORIGINAL nonzero root98 square := mul(98, 98)

After

Quadratic ORIGINAL nonzero root98 square = 9604

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S196

S198 · implies

Before

ORIGINAL nonzero root x=98 in1..100;actual square=9604

After

Euclidean square=95*101+9;nonzero square class=9;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S197

S199 · calculation

Before

Quadratic ORIGINAL nonzero root99 square := mul(99, 99)

After

Quadratic ORIGINAL nonzero root99 square = 9801

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S198

S200 · implies

Before

ORIGINAL nonzero root x=99 in1..100;actual square=9801

After

Euclidean square=97*101+4;nonzero square class=4;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S199

S201 · calculation

Before

Quadratic ORIGINAL nonzero root100 square := mul(100, 100)

After

Quadratic ORIGINAL nonzero root100 square = 10000

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S200

S202 · implies

Before

ORIGINAL nonzero root x=100 in1..100;actual square=10000

After

Euclidean square=99*101+1;nonzero square class=1;complete root-index order retained

Conditions: none

Enumerate every nonzero root with its exact square and modular decomposition.A nonzero x cannot square to0 modulo a prime.This is neither a residue sample nor an Euler/Legendre shortcut.

Step identifiers and dependencies

Rule: quadratic_ALL_nonzero_square_rows_not_sample · Uses: G001, S201

S203 · calculation

Before

Quadratic distinct nonzero residue count := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

After

Quadratic distinct nonzero residue count = 50

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S202

S204 · calculation

Before

Quadratic ORIGINAL modulus minus1 := sub(101, 1)

After

Quadratic ORIGINAL modulus minus1 = 100

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S203

S205 · calculation

Before

Quadratic half of all nonzero roots := div(100, 2)

After

Quadratic half of all nonzero roots = 50

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S204

S206 · implies

Before

ORIGINAL ODD PRIME p=101;complete nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 81], [10, 100], [11, 20], [12, 43], [13, 68], [14, 95], [15, 23], [16, 54], [17, 87], [18, 21], [19, 58], [20, 97], [21, 37], [22, 80], [23, 24], [24, 71], [25, 19], [26, 70], [27, 22], [28, 77], [29, 33], [30, 92], [31, 52], [32, 14], [33, 79], [34, 45], [35, 13], [36, 84], [37, 56], [38, 30], [39, 6], [40, 85], [41, 65], [42, 47], [43, 31], [44, 17], [45, 5], [46, 96], [47, 88], [48, 82], [49, 78], [50, 76], [51, 76], [52, 78], [53, 82], [54, 88], [55, 96], [56, 5], [57, 17], [58, 31], [59, 47], [60, 65], [61, 85], [62, 6], [63, 30], [64, 56], [65, 84], [66, 13], [67, 45], [68, 79], [69, 14], [70, 52], [71, 92], [72, 33], [73, 77], [74, 22], [75, 70], [76, 19], [77, 71], [78, 24], [79, 80], [80, 37], [81, 97], [82, 58], [83, 21], [84, 87], [85, 54], [86, 23], [87, 95], [88, 68], [89, 43], [90, 20], [91, 100], [92, 81], [93, 64], [94, 49], [95, 36], [96, 25], [97, 16], [98, 9], [99, 4], [100, 1]];distinct classes=[1, 4, 5, 6, 9, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, 30, 31, 33, 36, 37, 43, 45, 47, 49, 52, 54, 56, 58, 64, 65, 68, 70, 71, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, 88, 92, 95, 96, 97, 100]

After

x^2=y^2 implies p divides(x-y)(x+y),so x=y or x=-y modp;odd p and nonzero x make opposite roots DISTINCT;each image has2 roots,so residue count=50=(101-1)/2=50

Conditions: none

The prime-field no-zero-divisor property gives x=±y;oddness makes2x nonzero.The opposite pairs partition all p−1 nonzero roots.This proves both counting and two-root statements,not their extension to zero,2 or composite moduli.

Step identifiers and dependencies

Rule: quadratic_prime_field_opposite_root_pairs_exactly_two · Uses: G001, S205

S207 · calculation

Before

Quadratic ORIGINAL candidate class Euclidean quotient times modulus := mul(0, 101)

After

Quadratic ORIGINAL candidate class Euclidean quotient times modulus = 0

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S206

S208 · calculation

Before

Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple := sub(25, 0)

After

Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple = 25

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S207

S209 · implies

Before

Quadratic ORIGINAL candidate class:ORIGINAL signed value=25;positive modulus=101;actual quotient=0

After

canonical residue=25;value=quotient*modulus+residue and0<=residue<modulus

Conditions: none

Retain the original signed value and congruence class via Euclidean division;do not mistake a language-specific negative remainder for the canonical class.

Step identifiers and dependencies

Rule: signed_mod_ORIGINAL_class_normalized_not_changed · Uses: G001, S208

Reverse checks

S209 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S208 · premise_and_result_audit — Recompute Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple with exact rational arithmetic, not rounded decimals.
S207 · premise_and_result_audit — Recompute Quadratic ORIGINAL candidate class Euclidean quotient times modulus with exact rational arithmetic, not rounded decimals.
S206 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S205 · premise_and_result_audit — Recompute Quadratic half of all nonzero roots with exact rational arithmetic, not rounded decimals.
S204 · premise_and_result_audit — Recompute Quadratic ORIGINAL modulus minus1 with exact rational arithmetic, not rounded decimals.
S203 · premise_and_result_audit — Recompute Quadratic distinct nonzero residue count with exact rational arithmetic, not rounded decimals.
S202 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S201 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root100 square with exact rational arithmetic, not rounded decimals.
S200 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S199 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root99 square with exact rational arithmetic, not rounded decimals.
S198 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S197 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root98 square with exact rational arithmetic, not rounded decimals.
S196 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S195 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root97 square with exact rational arithmetic, not rounded decimals.
S194 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S193 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root96 square with exact rational arithmetic, not rounded decimals.
S192 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S191 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root95 square with exact rational arithmetic, not rounded decimals.
S190 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S189 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root94 square with exact rational arithmetic, not rounded decimals.
S188 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S187 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root93 square with exact rational arithmetic, not rounded decimals.
S186 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S185 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root92 square with exact rational arithmetic, not rounded decimals.
S184 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S183 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root91 square with exact rational arithmetic, not rounded decimals.
S182 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S181 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root90 square with exact rational arithmetic, not rounded decimals.
S180 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S179 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root89 square with exact rational arithmetic, not rounded decimals.
S178 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S177 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root88 square with exact rational arithmetic, not rounded decimals.
S176 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S175 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root87 square with exact rational arithmetic, not rounded decimals.
S174 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S173 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root86 square with exact rational arithmetic, not rounded decimals.
S172 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S171 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root85 square with exact rational arithmetic, not rounded decimals.
S170 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S169 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root84 square with exact rational arithmetic, not rounded decimals.
S168 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S167 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root83 square with exact rational arithmetic, not rounded decimals.
S166 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S165 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root82 square with exact rational arithmetic, not rounded decimals.
S164 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S163 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root81 square with exact rational arithmetic, not rounded decimals.
S162 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S161 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root80 square with exact rational arithmetic, not rounded decimals.
S160 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S159 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root79 square with exact rational arithmetic, not rounded decimals.
S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S157 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root78 square with exact rational arithmetic, not rounded decimals.
S156 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S155 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root77 square with exact rational arithmetic, not rounded decimals.
S154 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S153 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root76 square with exact rational arithmetic, not rounded decimals.
S152 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S151 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root75 square with exact rational arithmetic, not rounded decimals.
S150 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S149 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root74 square with exact rational arithmetic, not rounded decimals.
S148 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S147 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root73 square with exact rational arithmetic, not rounded decimals.
S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S145 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root72 square with exact rational arithmetic, not rounded decimals.
S144 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S143 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root71 square with exact rational arithmetic, not rounded decimals.
S142 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S141 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root70 square with exact rational arithmetic, not rounded decimals.
S140 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S139 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root69 square with exact rational arithmetic, not rounded decimals.
S138 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S137 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root68 square with exact rational arithmetic, not rounded decimals.
S136 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S135 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root67 square with exact rational arithmetic, not rounded decimals.
S134 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S133 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root66 square with exact rational arithmetic, not rounded decimals.
S132 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S131 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root65 square with exact rational arithmetic, not rounded decimals.
S130 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S129 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root64 square with exact rational arithmetic, not rounded decimals.
S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S127 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root63 square with exact rational arithmetic, not rounded decimals.
S126 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S125 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root62 square with exact rational arithmetic, not rounded decimals.
S124 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S123 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root61 square with exact rational arithmetic, not rounded decimals.
S122 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S121 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root60 square with exact rational arithmetic, not rounded decimals.
S120 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S119 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root59 square with exact rational arithmetic, not rounded decimals.
S118 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S117 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root58 square with exact rational arithmetic, not rounded decimals.
S116 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S115 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root57 square with exact rational arithmetic, not rounded decimals.
S114 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S113 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root56 square with exact rational arithmetic, not rounded decimals.
S112 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S111 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root55 square with exact rational arithmetic, not rounded decimals.
S110 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S109 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root54 square with exact rational arithmetic, not rounded decimals.
S108 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S107 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root53 square with exact rational arithmetic, not rounded decimals.
S106 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S105 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root52 square with exact rational arithmetic, not rounded decimals.
S104 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S103 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root51 square with exact rational arithmetic, not rounded decimals.
S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S101 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root50 square with exact rational arithmetic, not rounded decimals.
S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S099 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root49 square with exact rational arithmetic, not rounded decimals.
S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S097 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root48 square with exact rational arithmetic, not rounded decimals.
S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S095 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root47 square with exact rational arithmetic, not rounded decimals.
S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S093 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root46 square with exact rational arithmetic, not rounded decimals.
S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S091 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root45 square with exact rational arithmetic, not rounded decimals.
S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S089 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root44 square with exact rational arithmetic, not rounded decimals.
S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S087 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root43 square with exact rational arithmetic, not rounded decimals.
S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S085 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root42 square with exact rational arithmetic, not rounded decimals.
S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S083 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root41 square with exact rational arithmetic, not rounded decimals.
S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S081 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root40 square with exact rational arithmetic, not rounded decimals.
S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S079 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root39 square with exact rational arithmetic, not rounded decimals.
S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S077 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root38 square with exact rational arithmetic, not rounded decimals.
S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S075 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root37 square with exact rational arithmetic, not rounded decimals.
S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S073 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root36 square with exact rational arithmetic, not rounded decimals.
S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S071 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root35 square with exact rational arithmetic, not rounded decimals.
S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S069 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root34 square with exact rational arithmetic, not rounded decimals.
S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S067 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root33 square with exact rational arithmetic, not rounded decimals.
S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S065 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root32 square with exact rational arithmetic, not rounded decimals.
S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S063 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root31 square with exact rational arithmetic, not rounded decimals.
S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S061 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root30 square with exact rational arithmetic, not rounded decimals.
S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S059 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root29 square with exact rational arithmetic, not rounded decimals.
S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S057 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root28 square with exact rational arithmetic, not rounded decimals.
S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S055 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root27 square with exact rational arithmetic, not rounded decimals.
S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S053 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root26 square with exact rational arithmetic, not rounded decimals.
S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S051 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root25 square with exact rational arithmetic, not rounded decimals.
S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S049 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root24 square with exact rational arithmetic, not rounded decimals.
S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S047 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root23 square with exact rational arithmetic, not rounded decimals.
S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S045 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root22 square with exact rational arithmetic, not rounded decimals.
S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S043 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root21 square with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S041 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root20 square with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S039 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root19 square with exact rational arithmetic, not rounded decimals.
S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S037 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root18 square with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S035 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root17 square with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S033 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root16 square with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S031 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root15 square with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S029 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root14 square with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S027 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root13 square with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S025 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root12 square with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S023 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root11 square with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S021 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root10 square with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S019 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root9 square with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S017 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root8 square with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S015 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root7 square with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S013 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root6 square with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S011 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root5 square with exact rational arithmetic, not rounded decimals.
S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S009 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root4 square with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S007 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root3 square with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root2 square with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S003 · premise_and_result_audit — Recompute Quadratic ORIGINAL nonzero root1 square with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(101, 25, 25, [5, 96], Yes, nonzero_square_root_witness)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: a12c3eb8873652866b27fe2dd00a3c836e39b47c719dbf63797f91a84710ba4c
candidate record SHA-256: 8066b35af609b5b274f05f65268a9faaa94f5ba420cf344f1fcc6ab7d1e9264c
semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON