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Question 23710: 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

R0046-23710 · q-20261014-advance-20261112-r06-legendre-symbol-0010

Original question and givens

Use bounded odd prime p=31 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

  • G001 — Use bounded odd prime p=31 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=31;ALL possible trial divisors through square root=[2, 3, 4, 5]

After

ALL [divisor,remainder] rows=[[2, 1], [3, 1], [4, 3], [5, 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..30;exact square=1

After

square=0*31+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..30;exact square=4

After

square=0*31+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..30;exact square=9

After

square=0*31+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..30;exact square=16

After

square=0*31+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..30;exact square=25

After

square=0*31+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..30;exact square=36

After

square=1*31+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, 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..30;exact square=49

After

square=1*31+18;canonical square class=18;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..30;exact square=64

After

square=2*31+2;canonical square class=2;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..30;exact square=81

After

square=2*31+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, 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..30;exact square=100

After

square=3*31+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, 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..30;exact square=121

After

square=3*31+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, 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..30;exact square=144

After

square=4*31+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, 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..30;exact square=169

After

square=5*31+14;canonical square class=14;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..30;exact square=196

After

square=6*31+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, 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..30;exact square=225

After

square=7*31+8;canonical square class=8;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..30;exact square=256

After

square=8*31+8;canonical square class=8;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..30;exact square=289

After

square=9*31+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, 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..30;exact square=324

After

square=10*31+14;canonical square class=14;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..30;exact square=361

After

square=11*31+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, 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..30;exact square=400

After

square=12*31+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, 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..30;exact square=441

After

square=14*31+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, 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..30;exact square=484

After

square=15*31+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, 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..30;exact square=529

After

square=17*31+2;canonical square class=2;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..30;exact square=576

After

square=18*31+18;canonical square class=18;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..30;exact square=625

After

square=20*31+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, 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..30;exact square=676

After

square=21*31+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, 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..30;exact square=729

After

square=23*31+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, 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..30;exact square=784

After

square=25*31+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, 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..30;exact square=841

After

square=27*31+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, 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..30;exact square=900

After

square=29*31+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, S061

S063 · implies

Before

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

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, S062

Reverse checks

  • S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

(31, [[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]], 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: 18c3b2596d8db024363383ae45079be26c5d80e25e4101605e572425cb6cc20e
  • candidate record SHA-256: 7ae44d7c2121bece065114621bcb57599433c88d1859d4b597da302022280d1d
  • semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
  • question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON