Question 23711: Legendre Classes Include Zero before Nonzero Residue Classification

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

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R0046-23711 · q-20261014-advance-20261112-r06-legendre-symbol-0011

Original question and givens

Use bounded odd prime p=37 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

G001 — Use bounded odd prime p=37 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.

mko-legendre-symbol

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=37;ALL possible trial divisors through square root=[2, 3, 4, 5, 6]

After

ALL [divisor,remainder] rows=[[2, 1], [3, 1], [4, 1], [5, 2], [6, 1]];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..36;exact square=1

After

square=0*37+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..36;exact square=4

After

square=0*37+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..36;exact square=9

After

square=0*37+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..36;exact square=16

After

square=0*37+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..36;exact square=25

After

square=0*37+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..36;exact square=36

After

square=0*37+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..36;exact square=49

After

square=1*37+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, 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..36;exact square=64

After

square=1*37+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, 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..36;exact square=81

After

square=2*37+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, 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..36;exact square=100

After

square=2*37+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, 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..36;exact square=121

After

square=3*37+10;canonical square class=10;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..36;exact square=144

After

square=3*37+33;canonical square class=33;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..36;exact square=169

After

square=4*37+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, 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..36;exact square=196

After

square=5*37+11;canonical square class=11;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..36;exact square=225

After

square=6*37+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, 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..36;exact square=256

After

square=6*37+34;canonical square class=34;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..36;exact square=289

After

square=7*37+30;canonical square class=30;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..36;exact square=324

After

square=8*37+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, 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..36;exact square=361

After

square=9*37+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, 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..36;exact square=400

After

square=10*37+30;canonical square class=30;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..36;exact square=441

After

square=11*37+34;canonical square class=34;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..36;exact square=484

After

square=13*37+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, 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..36;exact square=529

After

square=14*37+11;canonical square class=11;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..36;exact square=576

After

square=15*37+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, 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..36;exact square=625

After

square=16*37+33;canonical square class=33;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..36;exact square=676

After

square=18*37+10;canonical square class=10;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..36;exact square=729

After

square=19*37+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, 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..36;exact square=784

After

square=21*37+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, 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..36;exact square=841

After

square=22*37+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, 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..36;exact square=900

After

square=24*37+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, 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..36;exact square=961

After

square=25*37+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, 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..36;exact square=1024

After

square=27*37+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, 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..36;exact square=1089

After

square=29*37+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, 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..36;exact square=1156

After

square=31*37+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, 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..36;exact square=1225

After

square=33*37+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, 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..36;exact square=1296

After

square=35*37+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, S073

S075 · implies

Before

ORIGINAL prime=37;COMPLETE nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 12], [8, 27], [9, 7], [10, 26], [11, 10], [12, 33], [13, 21], [14, 11], [15, 3], [16, 34], [17, 30], [18, 28], [19, 28], [20, 30], [21, 34], [22, 3], [23, 11], [24, 21], [25, 33], [26, 10], [27, 26], [28, 7], [29, 27], [30, 12], [31, 36], [32, 25], [33, 16], [34, 9], [35, 4], [36, 1]];nonzero square classes=[1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25, 26, 27, 28, 30, 33, 34, 36]

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, S074

Reverse checks

S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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

(37, [[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]], 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: 7eb63b1625c109407162d9a0e8ea9872283e06045d15a18a12c38bc3054820f1
candidate record SHA-256: 840948dfa07741a2e512426a95ac2c48d88387094b576ad20fc97603acd1251c
semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON