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Question 23716: Legendre Classes Include Zero before Nonzero Residue Classification

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

R0046-23716 · q-20261014-advance-20261112-r06-legendre-symbol-0016

Original question and givens

Use bounded odd prime p=59 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

  • G001 — Use bounded odd prime p=59 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, 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=legendre-classification-table;original snapshot and mathematics retained

After

explicit ordered source output fields=[prime, class_symbol_rows, complete_partition, certificate]

Conditions: none

Keep the source snapshot while listing ordered fields without JavaScript.Radius uses its original mode to select6 or11 fields without padding;distinguish nonclaims from actual negations.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

Legendre ORIGINAL odd prime:ORIGINAL modulus=59;ALL possible trial divisors through square root=[2, 3, 4, 5, 6, 7]

After

ALL [divisor,remainder] rows=[[2, 1], [3, 2], [4, 3], [5, 4], [6, 5], [7, 3]];prime=Yes

Conditions: none

Qualify primality over the full trial range,not by guessing from Legendre/Euler classification.This is a modulus premise,not the square status of a.

Step identifiers and dependencies

Rule: legendre_complete_prime_trial_not_Euler_classification · Uses: G001, S001

S003 · calculation

Before

Legendre ORIGINAL nonzero root1 exact square := mul(1, 1)

After

Legendre ORIGINAL nonzero root1 exact square = 1

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

ORIGINAL root x=1 in1..58;exact square=1

After

square=0*59+1;canonical square class=1;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S003

S005 · calculation

Before

Legendre ORIGINAL nonzero root2 exact square := mul(2, 2)

After

Legendre ORIGINAL nonzero root2 exact square = 4

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

ORIGINAL root x=2 in1..58;exact square=4

After

square=0*59+4;canonical square class=4;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S005

S007 · calculation

Before

Legendre ORIGINAL nonzero root3 exact square := mul(3, 3)

After

Legendre ORIGINAL nonzero root3 exact square = 9

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

ORIGINAL root x=3 in1..58;exact square=9

After

square=0*59+9;canonical square class=9;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S007

S009 · calculation

Before

Legendre ORIGINAL nonzero root4 exact square := mul(4, 4)

After

Legendre ORIGINAL nonzero root4 exact square = 16

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

ORIGINAL root x=4 in1..58;exact square=16

After

square=0*59+16;canonical square class=16;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S009

S011 · calculation

Before

Legendre ORIGINAL nonzero root5 exact square := mul(5, 5)

After

Legendre ORIGINAL nonzero root5 exact square = 25

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

ORIGINAL root x=5 in1..58;exact square=25

After

square=0*59+25;canonical square class=25;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S011

S013 · calculation

Before

Legendre ORIGINAL nonzero root6 exact square := mul(6, 6)

After

Legendre ORIGINAL nonzero root6 exact square = 36

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

ORIGINAL root x=6 in1..58;exact square=36

After

square=0*59+36;canonical square class=36;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S013

S015 · calculation

Before

Legendre ORIGINAL nonzero root7 exact square := mul(7, 7)

After

Legendre ORIGINAL nonzero root7 exact square = 49

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

ORIGINAL root x=7 in1..58;exact square=49

After

square=0*59+49;canonical square class=49;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S015

S017 · calculation

Before

Legendre ORIGINAL nonzero root8 exact square := mul(8, 8)

After

Legendre ORIGINAL nonzero root8 exact square = 64

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

ORIGINAL root x=8 in1..58;exact square=64

After

square=1*59+5;canonical square class=5;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S017

S019 · calculation

Before

Legendre ORIGINAL nonzero root9 exact square := mul(9, 9)

After

Legendre ORIGINAL nonzero root9 exact square = 81

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

ORIGINAL root x=9 in1..58;exact square=81

After

square=1*59+22;canonical square class=22;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S019

S021 · calculation

Before

Legendre ORIGINAL nonzero root10 exact square := mul(10, 10)

After

Legendre ORIGINAL nonzero root10 exact square = 100

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

ORIGINAL root x=10 in1..58;exact square=100

After

square=1*59+41;canonical square class=41;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S021

S023 · calculation

Before

Legendre ORIGINAL nonzero root11 exact square := mul(11, 11)

After

Legendre ORIGINAL nonzero root11 exact square = 121

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

ORIGINAL root x=11 in1..58;exact square=121

After

square=2*59+3;canonical square class=3;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S023

S025 · calculation

Before

Legendre ORIGINAL nonzero root12 exact square := mul(12, 12)

After

Legendre ORIGINAL nonzero root12 exact square = 144

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

ORIGINAL root x=12 in1..58;exact square=144

After

square=2*59+26;canonical square class=26;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S025

S027 · calculation

Before

Legendre ORIGINAL nonzero root13 exact square := mul(13, 13)

After

Legendre ORIGINAL nonzero root13 exact square = 169

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

ORIGINAL root x=13 in1..58;exact square=169

After

square=2*59+51;canonical square class=51;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S027

S029 · calculation

Before

Legendre ORIGINAL nonzero root14 exact square := mul(14, 14)

After

Legendre ORIGINAL nonzero root14 exact square = 196

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · implies

Before

ORIGINAL root x=14 in1..58;exact square=196

After

square=3*59+19;canonical square class=19;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S029

S031 · calculation

Before

Legendre ORIGINAL nonzero root15 exact square := mul(15, 15)

After

Legendre ORIGINAL nonzero root15 exact square = 225

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · implies

Before

ORIGINAL root x=15 in1..58;exact square=225

After

square=3*59+48;canonical square class=48;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S031

S033 · calculation

Before

Legendre ORIGINAL nonzero root16 exact square := mul(16, 16)

After

Legendre ORIGINAL nonzero root16 exact square = 256

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · implies

Before

ORIGINAL root x=16 in1..58;exact square=256

After

square=4*59+20;canonical square class=20;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S033

S035 · calculation

Before

Legendre ORIGINAL nonzero root17 exact square := mul(17, 17)

After

Legendre ORIGINAL nonzero root17 exact square = 289

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · implies

Before

ORIGINAL root x=17 in1..58;exact square=289

After

square=4*59+53;canonical square class=53;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S035

S037 · calculation

Before

Legendre ORIGINAL nonzero root18 exact square := mul(18, 18)

After

Legendre ORIGINAL nonzero root18 exact square = 324

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

ORIGINAL root x=18 in1..58;exact square=324

After

square=5*59+29;canonical square class=29;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S037

S039 · calculation

Before

Legendre ORIGINAL nonzero root19 exact square := mul(19, 19)

After

Legendre ORIGINAL nonzero root19 exact square = 361

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · implies

Before

ORIGINAL root x=19 in1..58;exact square=361

After

square=6*59+7;canonical square class=7;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S039

S041 · calculation

Before

Legendre ORIGINAL nonzero root20 exact square := mul(20, 20)

After

Legendre ORIGINAL nonzero root20 exact square = 400

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · implies

Before

ORIGINAL root x=20 in1..58;exact square=400

After

square=6*59+46;canonical square class=46;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S041

S043 · calculation

Before

Legendre ORIGINAL nonzero root21 exact square := mul(21, 21)

After

Legendre ORIGINAL nonzero root21 exact square = 441

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · implies

Before

ORIGINAL root x=21 in1..58;exact square=441

After

square=7*59+28;canonical square class=28;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S043

S045 · calculation

Before

Legendre ORIGINAL nonzero root22 exact square := mul(22, 22)

After

Legendre ORIGINAL nonzero root22 exact square = 484

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · implies

Before

ORIGINAL root x=22 in1..58;exact square=484

After

square=8*59+12;canonical square class=12;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S045

S047 · calculation

Before

Legendre ORIGINAL nonzero root23 exact square := mul(23, 23)

After

Legendre ORIGINAL nonzero root23 exact square = 529

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · implies

Before

ORIGINAL root x=23 in1..58;exact square=529

After

square=8*59+57;canonical square class=57;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S047

S049 · calculation

Before

Legendre ORIGINAL nonzero root24 exact square := mul(24, 24)

After

Legendre ORIGINAL nonzero root24 exact square = 576

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · implies

Before

ORIGINAL root x=24 in1..58;exact square=576

After

square=9*59+45;canonical square class=45;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S049

S051 · calculation

Before

Legendre ORIGINAL nonzero root25 exact square := mul(25, 25)

After

Legendre ORIGINAL nonzero root25 exact square = 625

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · implies

Before

ORIGINAL root x=25 in1..58;exact square=625

After

square=10*59+35;canonical square class=35;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S051

S053 · calculation

Before

Legendre ORIGINAL nonzero root26 exact square := mul(26, 26)

After

Legendre ORIGINAL nonzero root26 exact square = 676

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · implies

Before

ORIGINAL root x=26 in1..58;exact square=676

After

square=11*59+27;canonical square class=27;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S053

S055 · calculation

Before

Legendre ORIGINAL nonzero root27 exact square := mul(27, 27)

After

Legendre ORIGINAL nonzero root27 exact square = 729

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · implies

Before

ORIGINAL root x=27 in1..58;exact square=729

After

square=12*59+21;canonical square class=21;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S055

S057 · calculation

Before

Legendre ORIGINAL nonzero root28 exact square := mul(28, 28)

After

Legendre ORIGINAL nonzero root28 exact square = 784

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · implies

Before

ORIGINAL root x=28 in1..58;exact square=784

After

square=13*59+17;canonical square class=17;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S057

S059 · calculation

Before

Legendre ORIGINAL nonzero root29 exact square := mul(29, 29)

After

Legendre ORIGINAL nonzero root29 exact square = 841

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · implies

Before

ORIGINAL root x=29 in1..58;exact square=841

After

square=14*59+15;canonical square class=15;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S059

S061 · calculation

Before

Legendre ORIGINAL nonzero root30 exact square := mul(30, 30)

After

Legendre ORIGINAL nonzero root30 exact square = 900

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · implies

Before

ORIGINAL root x=30 in1..58;exact square=900

After

square=15*59+15;canonical square class=15;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S061

S063 · calculation

Before

Legendre ORIGINAL nonzero root31 exact square := mul(31, 31)

After

Legendre ORIGINAL nonzero root31 exact square = 961

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · implies

Before

ORIGINAL root x=31 in1..58;exact square=961

After

square=16*59+17;canonical square class=17;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S063

S065 · calculation

Before

Legendre ORIGINAL nonzero root32 exact square := mul(32, 32)

After

Legendre ORIGINAL nonzero root32 exact square = 1024

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · implies

Before

ORIGINAL root x=32 in1..58;exact square=1024

After

square=17*59+21;canonical square class=21;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S065

S067 · calculation

Before

Legendre ORIGINAL nonzero root33 exact square := mul(33, 33)

After

Legendre ORIGINAL nonzero root33 exact square = 1089

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · implies

Before

ORIGINAL root x=33 in1..58;exact square=1089

After

square=18*59+27;canonical square class=27;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S067

S069 · calculation

Before

Legendre ORIGINAL nonzero root34 exact square := mul(34, 34)

After

Legendre ORIGINAL nonzero root34 exact square = 1156

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · implies

Before

ORIGINAL root x=34 in1..58;exact square=1156

After

square=19*59+35;canonical square class=35;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S069

S071 · calculation

Before

Legendre ORIGINAL nonzero root35 exact square := mul(35, 35)

After

Legendre ORIGINAL nonzero root35 exact square = 1225

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · implies

Before

ORIGINAL root x=35 in1..58;exact square=1225

After

square=20*59+45;canonical square class=45;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S071

S073 · calculation

Before

Legendre ORIGINAL nonzero root36 exact square := mul(36, 36)

After

Legendre ORIGINAL nonzero root36 exact square = 1296

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · implies

Before

ORIGINAL root x=36 in1..58;exact square=1296

After

square=21*59+57;canonical square class=57;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S073

S075 · calculation

Before

Legendre ORIGINAL nonzero root37 exact square := mul(37, 37)

After

Legendre ORIGINAL nonzero root37 exact square = 1369

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · implies

Before

ORIGINAL root x=37 in1..58;exact square=1369

After

square=23*59+12;canonical square class=12;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S075

S077 · calculation

Before

Legendre ORIGINAL nonzero root38 exact square := mul(38, 38)

After

Legendre ORIGINAL nonzero root38 exact square = 1444

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S076

S078 · implies

Before

ORIGINAL root x=38 in1..58;exact square=1444

After

square=24*59+28;canonical square class=28;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S077

S079 · calculation

Before

Legendre ORIGINAL nonzero root39 exact square := mul(39, 39)

After

Legendre ORIGINAL nonzero root39 exact square = 1521

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S078

S080 · implies

Before

ORIGINAL root x=39 in1..58;exact square=1521

After

square=25*59+46;canonical square class=46;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S079

S081 · calculation

Before

Legendre ORIGINAL nonzero root40 exact square := mul(40, 40)

After

Legendre ORIGINAL nonzero root40 exact square = 1600

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S080

S082 · implies

Before

ORIGINAL root x=40 in1..58;exact square=1600

After

square=27*59+7;canonical square class=7;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S081

S083 · calculation

Before

Legendre ORIGINAL nonzero root41 exact square := mul(41, 41)

After

Legendre ORIGINAL nonzero root41 exact square = 1681

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S082

S084 · implies

Before

ORIGINAL root x=41 in1..58;exact square=1681

After

square=28*59+29;canonical square class=29;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S083

S085 · calculation

Before

Legendre ORIGINAL nonzero root42 exact square := mul(42, 42)

After

Legendre ORIGINAL nonzero root42 exact square = 1764

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S084

S086 · implies

Before

ORIGINAL root x=42 in1..58;exact square=1764

After

square=29*59+53;canonical square class=53;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S085

S087 · calculation

Before

Legendre ORIGINAL nonzero root43 exact square := mul(43, 43)

After

Legendre ORIGINAL nonzero root43 exact square = 1849

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S086

S088 · implies

Before

ORIGINAL root x=43 in1..58;exact square=1849

After

square=31*59+20;canonical square class=20;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S087

S089 · calculation

Before

Legendre ORIGINAL nonzero root44 exact square := mul(44, 44)

After

Legendre ORIGINAL nonzero root44 exact square = 1936

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S088

S090 · implies

Before

ORIGINAL root x=44 in1..58;exact square=1936

After

square=32*59+48;canonical square class=48;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S089

S091 · calculation

Before

Legendre ORIGINAL nonzero root45 exact square := mul(45, 45)

After

Legendre ORIGINAL nonzero root45 exact square = 2025

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S090

S092 · implies

Before

ORIGINAL root x=45 in1..58;exact square=2025

After

square=34*59+19;canonical square class=19;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S091

S093 · calculation

Before

Legendre ORIGINAL nonzero root46 exact square := mul(46, 46)

After

Legendre ORIGINAL nonzero root46 exact square = 2116

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S092

S094 · implies

Before

ORIGINAL root x=46 in1..58;exact square=2116

After

square=35*59+51;canonical square class=51;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S093

S095 · calculation

Before

Legendre ORIGINAL nonzero root47 exact square := mul(47, 47)

After

Legendre ORIGINAL nonzero root47 exact square = 2209

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S094

S096 · implies

Before

ORIGINAL root x=47 in1..58;exact square=2209

After

square=37*59+26;canonical square class=26;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S095

S097 · calculation

Before

Legendre ORIGINAL nonzero root48 exact square := mul(48, 48)

After

Legendre ORIGINAL nonzero root48 exact square = 2304

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S096

S098 · implies

Before

ORIGINAL root x=48 in1..58;exact square=2304

After

square=39*59+3;canonical square class=3;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S097

S099 · calculation

Before

Legendre ORIGINAL nonzero root49 exact square := mul(49, 49)

After

Legendre ORIGINAL nonzero root49 exact square = 2401

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S098

S100 · implies

Before

ORIGINAL root x=49 in1..58;exact square=2401

After

square=40*59+41;canonical square class=41;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S099

S101 · calculation

Before

Legendre ORIGINAL nonzero root50 exact square := mul(50, 50)

After

Legendre ORIGINAL nonzero root50 exact square = 2500

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S100

S102 · implies

Before

ORIGINAL root x=50 in1..58;exact square=2500

After

square=42*59+22;canonical square class=22;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S101

S103 · calculation

Before

Legendre ORIGINAL nonzero root51 exact square := mul(51, 51)

After

Legendre ORIGINAL nonzero root51 exact square = 2601

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S102

S104 · implies

Before

ORIGINAL root x=51 in1..58;exact square=2601

After

square=44*59+5;canonical square class=5;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S103

S105 · calculation

Before

Legendre ORIGINAL nonzero root52 exact square := mul(52, 52)

After

Legendre ORIGINAL nonzero root52 exact square = 2704

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S104

S106 · implies

Before

ORIGINAL root x=52 in1..58;exact square=2704

After

square=45*59+49;canonical square class=49;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S105

S107 · calculation

Before

Legendre ORIGINAL nonzero root53 exact square := mul(53, 53)

After

Legendre ORIGINAL nonzero root53 exact square = 2809

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S106

S108 · implies

Before

ORIGINAL root x=53 in1..58;exact square=2809

After

square=47*59+36;canonical square class=36;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S107

S109 · calculation

Before

Legendre ORIGINAL nonzero root54 exact square := mul(54, 54)

After

Legendre ORIGINAL nonzero root54 exact square = 2916

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S108

S110 · implies

Before

ORIGINAL root x=54 in1..58;exact square=2916

After

square=49*59+25;canonical square class=25;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S109

S111 · calculation

Before

Legendre ORIGINAL nonzero root55 exact square := mul(55, 55)

After

Legendre ORIGINAL nonzero root55 exact square = 3025

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S110

S112 · implies

Before

ORIGINAL root x=55 in1..58;exact square=3025

After

square=51*59+16;canonical square class=16;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S111

S113 · calculation

Before

Legendre ORIGINAL nonzero root56 exact square := mul(56, 56)

After

Legendre ORIGINAL nonzero root56 exact square = 3136

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S112

S114 · implies

Before

ORIGINAL root x=56 in1..58;exact square=3136

After

square=53*59+9;canonical square class=9;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S113

S115 · calculation

Before

Legendre ORIGINAL nonzero root57 exact square := mul(57, 57)

After

Legendre ORIGINAL nonzero root57 exact square = 3249

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S114

S116 · implies

Before

ORIGINAL root x=57 in1..58;exact square=3249

After

square=55*59+4;canonical square class=4;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S115

S117 · calculation

Before

Legendre ORIGINAL nonzero root58 exact square := mul(58, 58)

After

Legendre ORIGINAL nonzero root58 exact square = 3364

Conditions: none

Substitute original roles and established intermediates exactly,without floats or treating finite tests as universal limits.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S116

S118 · implies

Before

ORIGINAL root x=58 in1..58;exact square=3364

After

square=57*59+1;canonical square class=1;ALL root indices retained

Conditions: none

Include every nonzero root.A few squares do not prove another class rootless;no Euler criterion is pre-used.

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, S117

S119 · implies

Before

ORIGINAL prime=59;COMPLETE nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 5], [9, 22], [10, 41], [11, 3], [12, 26], [13, 51], [14, 19], [15, 48], [16, 20], [17, 53], [18, 29], [19, 7], [20, 46], [21, 28], [22, 12], [23, 57], [24, 45], [25, 35], [26, 27], [27, 21], [28, 17], [29, 15], [30, 15], [31, 17], [32, 21], [33, 27], [34, 35], [35, 45], [36, 57], [37, 12], [38, 28], [39, 46], [40, 7], [41, 29], [42, 53], [43, 20], [44, 48], [45, 19], [46, 51], [47, 26], [48, 3], [49, 41], [50, 22], [51, 5], [52, 49], [53, 36], [54, 25], [55, 16], [56, 9], [57, 4], [58, 1]];nonzero square classes=[1, 3, 4, 5, 7, 9, 12, 15, 16, 17, 19, 20, 21, 22, 25, 26, 27, 28, 29, 35, 36, 41, 45, 46, 48, 49, 51, 53, 57]

After

for EACH normalized class r,first r0 ->symbol0;else square-table member ->symbol1;otherwise ->symbol-1;these cases partition ALL0..p-1

Conditions: none

Separate zero first;the fact that0 is a square does not give symbol+1.Nonzero classes are classified by the complete square table.This is definition plus exhaustive evidence,not a later multiplicativity or Euler formula.

Step identifiers and dependencies

Rule: legendre_ZERO_then_residue_nonresidue_three_way_definition · Uses: G001, S118

Reverse checks

  • S119 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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 Legendre ORIGINAL nonzero root58 exact 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 Legendre ORIGINAL nonzero root57 exact 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 Legendre ORIGINAL nonzero root56 exact 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 Legendre ORIGINAL nonzero root55 exact 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 Legendre ORIGINAL nonzero root54 exact 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 Legendre ORIGINAL nonzero root53 exact 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 Legendre ORIGINAL nonzero root52 exact 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 Legendre ORIGINAL nonzero root51 exact 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 Legendre ORIGINAL nonzero root50 exact 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 Legendre ORIGINAL nonzero root49 exact 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 Legendre ORIGINAL nonzero root48 exact 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 Legendre ORIGINAL nonzero root47 exact 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 Legendre ORIGINAL nonzero root46 exact 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 Legendre ORIGINAL nonzero root45 exact 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 Legendre ORIGINAL nonzero root44 exact 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 Legendre ORIGINAL nonzero root43 exact 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 Legendre ORIGINAL nonzero root42 exact 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 Legendre ORIGINAL nonzero root41 exact 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 Legendre ORIGINAL nonzero root40 exact 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 Legendre ORIGINAL nonzero root39 exact 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 Legendre ORIGINAL nonzero root38 exact 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 Legendre ORIGINAL nonzero root37 exact 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 Legendre ORIGINAL nonzero root36 exact 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 Legendre ORIGINAL nonzero root35 exact 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 Legendre ORIGINAL nonzero root34 exact 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 Legendre ORIGINAL nonzero root33 exact 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 Legendre ORIGINAL nonzero root32 exact 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 Legendre ORIGINAL nonzero root31 exact 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 Legendre ORIGINAL nonzero root30 exact 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 Legendre ORIGINAL nonzero root29 exact 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 Legendre ORIGINAL nonzero root28 exact 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 Legendre ORIGINAL nonzero root27 exact 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 Legendre ORIGINAL nonzero root26 exact 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 Legendre ORIGINAL nonzero root25 exact 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 Legendre ORIGINAL nonzero root24 exact 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 Legendre ORIGINAL nonzero root23 exact 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 Legendre ORIGINAL nonzero root22 exact 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 Legendre ORIGINAL nonzero root21 exact 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 Legendre ORIGINAL nonzero root20 exact 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 Legendre ORIGINAL nonzero root19 exact 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 Legendre ORIGINAL nonzero root18 exact 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 Legendre ORIGINAL nonzero root17 exact 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 Legendre ORIGINAL nonzero root16 exact 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 Legendre ORIGINAL nonzero root15 exact 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 Legendre ORIGINAL nonzero root14 exact 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 Legendre ORIGINAL nonzero root13 exact 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 Legendre ORIGINAL nonzero root12 exact 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 Legendre ORIGINAL nonzero root11 exact 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 Legendre ORIGINAL nonzero root10 exact 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 Legendre ORIGINAL nonzero root9 exact 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 Legendre ORIGINAL nonzero root8 exact 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 Legendre ORIGINAL nonzero root7 exact 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 Legendre ORIGINAL nonzero root6 exact 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 Legendre ORIGINAL nonzero root5 exact 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 Legendre ORIGINAL nonzero root4 exact 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 Legendre ORIGINAL nonzero root3 exact 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 Legendre ORIGINAL nonzero root2 exact 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 Legendre ORIGINAL nonzero root1 exact 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

(59, [[0, 0], [1, 1], [2, -1], [3, 1], [4, 1], [5, 1], [6, -1], [7, 1], [8, -1], [9, 1], [10, -1], [11, -1], [12, 1], [13, -1], [14, -1], [15, 1], [16, 1], [17, 1], [18, -1], [19, 1], [20, 1], [21, 1], [22, 1], [23, -1], [24, -1], [25, 1], [26, 1], [27, 1], [28, 1], [29, 1], [30, -1], [31, -1], [32, -1], [33, -1], [34, -1], [35, 1], [36, 1], [37, -1], [38, -1], [39, -1], [40, -1], [41, 1], [42, -1], [43, -1], [44, -1], [45, 1], [46, 1], [47, -1], [48, 1], [49, 1], [50, -1], [51, 1], [52, -1], [53, 1], [54, -1], [55, -1], [56, -1], [57, 1], [58, -1]], Yes, zero_residue_nonresidue_partition_is_complete)

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: b1bba3140289430d1654ee2713d58cbe8972b394a820c6628fb7543a35ee38c8
  • candidate record SHA-256: 49f52162748e7e7a135d006b87c33cd1996222a901a3602a06dcfd18fc3dec21
  • semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
  • question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON