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Question 23705: 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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R0046-23705 · q-20261014-advance-20261112-r06-legendre-symbol-0005

Original question and givens

Use bounded odd prime p=13 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

  • G001 — Use bounded odd prime p=13 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=13;ALL possible trial divisors through square root=[2, 3]

After

ALL [divisor,remainder] rows=[[2, 1], [3, 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..12;exact square=1

After

square=0*13+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..12;exact square=4

After

square=0*13+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..12;exact square=9

After

square=0*13+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..12;exact square=16

After

square=1*13+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, 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..12;exact square=25

After

square=1*13+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, 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..12;exact square=36

After

square=2*13+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, 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..12;exact square=49

After

square=3*13+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, 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..12;exact square=64

After

square=4*13+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, 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..12;exact square=81

After

square=6*13+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, 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..12;exact square=100

After

square=7*13+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, 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..12;exact square=121

After

square=9*13+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, 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..12;exact square=144

After

square=11*13+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, S025

S027 · implies

Before

ORIGINAL prime=13;COMPLETE nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 3], [5, 12], [6, 10], [7, 10], [8, 12], [9, 3], [10, 9], [11, 4], [12, 1]];nonzero square classes=[1, 3, 4, 9, 10, 12]

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, S026

Reverse checks

  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

(13, [[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]], 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: 647986f599b3703cab0945f140e5b25dff3fa48f4046622ab956fa6dd6605e84
  • candidate record SHA-256: d65fbc5cd952177e35fff90d14024efbcb211a5067e6fbe7eae501e0950bb5fb
  • semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
  • question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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