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Question 23365: A Nonzero Residue Modulo an Odd Prime Has Two Distinct Opposite Roots

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0045-23365 · q-20261014-advance-20261111-r06-quadratic-residue-0065

Original question and givens

Use bounded odd prime p=53 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=13. Give (p, normalized_class, roots, opposite_first_root, opposite_present, distinct_roots, certificate).

  • G001 — Use bounded odd prime p=53 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=13. Give (p, normalized_class, roots, opposite_first_root, opposite_present, distinct_roots, 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.

Step-by-step derivation

S001 · implies

Before

source operation=quadratic-paired-roots;original snapshot and mathematics retained

After

explicit ordered source output fields=[p, normalized_class, roots, opposite_first_root, opposite_present, distinct_roots, 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=53;complete possible trial divisors=[2, 3, 4, 5, 6, 7]

After

ALL [divisor,remainder] rows=[[2, 1], [3, 2], [4, 1], [5, 3], [6, 5], [7, 4]];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..52;actual square=1

After

Euclidean square=0*53+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..52;actual square=4

After

Euclidean square=0*53+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..52;actual square=9

After

Euclidean square=0*53+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..52;actual square=16

After

Euclidean square=0*53+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..52;actual square=25

After

Euclidean square=0*53+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..52;actual square=36

After

Euclidean square=0*53+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..52;actual square=49

After

Euclidean square=0*53+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..52;actual square=64

After

Euclidean square=1*53+11;nonzero square class=11;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..52;actual square=81

After

Euclidean square=1*53+28;nonzero square class=28;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..52;actual square=100

After

Euclidean square=1*53+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, 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..52;actual square=121

After

Euclidean square=2*53+15;nonzero square class=15;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..52;actual square=144

After

Euclidean square=2*53+38;nonzero square class=38;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..52;actual square=169

After

Euclidean square=3*53+10;nonzero square class=10;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..52;actual square=196

After

Euclidean square=3*53+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, 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..52;actual square=225

After

Euclidean square=4*53+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, 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..52;actual square=256

After

Euclidean square=4*53+44;nonzero square class=44;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..52;actual square=289

After

Euclidean square=5*53+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, 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..52;actual square=324

After

Euclidean square=6*53+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, 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..52;actual square=361

After

Euclidean square=6*53+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, 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..52;actual square=400

After

Euclidean square=7*53+29;nonzero square class=29;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..52;actual square=441

After

Euclidean square=8*53+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, 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..52;actual square=484

After

Euclidean square=9*53+7;nonzero square class=7;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..52;actual square=529

After

Euclidean square=9*53+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, 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..52;actual square=576

After

Euclidean square=10*53+46;nonzero square class=46;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..52;actual square=625

After

Euclidean square=11*53+42;nonzero square class=42;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..52;actual square=676

After

Euclidean square=12*53+40;nonzero square class=40;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..52;actual square=729

After

Euclidean square=13*53+40;nonzero square class=40;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..52;actual square=784

After

Euclidean square=14*53+42;nonzero square class=42;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..52;actual square=841

After

Euclidean square=15*53+46;nonzero square class=46;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..52;actual square=900

After

Euclidean square=16*53+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, 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..52;actual square=961

After

Euclidean square=18*53+7;nonzero square class=7;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..52;actual square=1024

After

Euclidean square=19*53+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, 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..52;actual square=1089

After

Euclidean square=20*53+29;nonzero square class=29;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..52;actual square=1156

After

Euclidean square=21*53+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, 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..52;actual square=1225

After

Euclidean square=23*53+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, 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..52;actual square=1296

After

Euclidean square=24*53+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, 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..52;actual square=1369

After

Euclidean square=25*53+44;nonzero square class=44;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..52;actual square=1444

After

Euclidean square=27*53+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, 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..52;actual square=1521

After

Euclidean square=28*53+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, 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..52;actual square=1600

After

Euclidean square=30*53+10;nonzero square class=10;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..52;actual square=1681

After

Euclidean square=31*53+38;nonzero square class=38;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..52;actual square=1764

After

Euclidean square=33*53+15;nonzero square class=15;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..52;actual square=1849

After

Euclidean square=34*53+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, 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..52;actual square=1936

After

Euclidean square=36*53+28;nonzero square class=28;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..52;actual square=2025

After

Euclidean square=38*53+11;nonzero square class=11;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..52;actual square=2116

After

Euclidean square=39*53+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, 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..52;actual square=2209

After

Euclidean square=41*53+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, 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..52;actual square=2304

After

Euclidean square=43*53+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, 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..52;actual square=2401

After

Euclidean square=45*53+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, 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..52;actual square=2500

After

Euclidean square=47*53+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, 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..52;actual square=2601

After

Euclidean square=49*53+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, 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..52;actual square=2704

After

Euclidean square=51*53+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, S105

S107 · 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)

After

Quadratic distinct nonzero residue count = 26

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 · calculation

Before

Quadratic ORIGINAL modulus minus1 := sub(53, 1)

After

Quadratic ORIGINAL modulus minus1 = 52

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, S107

S109 · calculation

Before

Quadratic half of all nonzero roots := div(52, 2)

After

Quadratic half of all nonzero roots = 26

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 ODD PRIME p=53;complete nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 11], [9, 28], [10, 47], [11, 15], [12, 38], [13, 10], [14, 37], [15, 13], [16, 44], [17, 24], [18, 6], [19, 43], [20, 29], [21, 17], [22, 7], [23, 52], [24, 46], [25, 42], [26, 40], [27, 40], [28, 42], [29, 46], [30, 52], [31, 7], [32, 17], [33, 29], [34, 43], [35, 6], [36, 24], [37, 44], [38, 13], [39, 37], [40, 10], [41, 38], [42, 15], [43, 47], [44, 28], [45, 11], [46, 49], [47, 36], [48, 25], [49, 16], [50, 9], [51, 4], [52, 1]];distinct classes=[1, 4, 6, 7, 9, 10, 11, 13, 15, 16, 17, 24, 25, 28, 29, 36, 37, 38, 40, 42, 43, 44, 46, 47, 49, 52]

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=26=(53-1)/2=26

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, S109

S111 · calculation

Before

Quadratic ORIGINAL candidate class Euclidean quotient times modulus := mul(0, 53)

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, S110

S112 · calculation

Before

Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple := sub(13, 0)

After

Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple = 13

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, S111

S113 · implies

Before

Quadratic ORIGINAL candidate class:ORIGINAL signed value=13;positive modulus=53;actual quotient=0

After

canonical residue=13;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, S112

S114 · calculation

Before

Quadratic negative of FIRST source root Euclidean quotient times modulus := mul(-1, 53)

After

Quadratic negative of FIRST source root Euclidean quotient times modulus = -53

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, S113

S115 · calculation

Before

Quadratic negative of FIRST source root ORIGINAL signed value minus multiple := sub(-15, -53)

After

Quadratic negative of FIRST source root ORIGINAL signed value minus multiple = 38

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

Quadratic negative of FIRST source root:ORIGINAL signed value=-15;positive modulus=53;actual quotient=-1

After

canonical residue=38;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, S115

Reverse checks

  • 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 negative of FIRST source root ORIGINAL signed value minus multiple with exact rational arithmetic, not rounded decimals.
  • S114 · premise_and_result_audit — Recompute Quadratic negative of FIRST source root Euclidean quotient times modulus with exact rational arithmetic, not rounded decimals.
  • S113 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S112 · premise_and_result_audit — Recompute Quadratic ORIGINAL candidate class ORIGINAL signed value minus multiple with exact rational arithmetic, not rounded decimals.
  • S111 · premise_and_result_audit — Recompute Quadratic ORIGINAL candidate class Euclidean quotient times modulus 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 half of all nonzero roots with exact rational arithmetic, not rounded decimals.
  • S108 · premise_and_result_audit — Recompute Quadratic ORIGINAL modulus minus1 with exact rational arithmetic, not rounded decimals.
  • S107 · premise_and_result_audit — Recompute Quadratic distinct nonzero residue count 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

(53, 13, [15, 38], 38, Yes, Yes, odd_prime_nonzero_residue_has_two_distinct_opposite_roots)

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: 65cc17f2936a5d001de0ca41f1ac9f890e1091f5368b4318402fc21afbb0d5e8
  • candidate record SHA-256: 6f9580e9fd2f117b784dc900cdca63d0898ca9b1ad508aa405bd920ff31e82d7
  • semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
  • question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON