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Question 23757: A Complete Root Table Distinguishes Symbol Plus One from Minus One

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

R0046-23757 · q-20261014-advance-20261112-r06-legendre-symbol-0032

Original question and givens

Use odd prime p=19, original a=7. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, certificate).

  • G001 — Use odd prime p=19, original a=7. Give (prime, a, normalized_class, nonzero_roots, Legendre_symbol, 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-residue-nonresidue-witness;original snapshot and mathematics retained

After

explicit ordered source output fields=[prime, a, normalized_class, nonzero_roots, Legendre_symbol, 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=19;ALL possible trial divisors through square root=[2, 3, 4]

After

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

Conditions: none

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

Step identifiers and dependencies

Rule: legendre_complete_prime_trial_not_Euler_classification · Uses: G001, S001

S003 · calculation

Before

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

After

Legendre ORIGINAL nonzero root1 exact square = 1

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

ORIGINAL root x=1 in1..18;exact square=1

After

square=0*19+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..18;exact square=4

After

square=0*19+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..18;exact square=9

After

square=0*19+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..18;exact square=16

After

square=0*19+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..18;exact square=25

After

square=1*19+6;canonical square class=6;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..18;exact square=36

After

square=1*19+17;canonical square class=17;ALL root indices retained

Conditions: none

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

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, 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..18;exact square=49

After

square=2*19+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, 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..18;exact square=64

After

square=3*19+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, 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..18;exact square=81

After

square=4*19+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, 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..18;exact square=100

After

square=5*19+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, 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..18;exact square=121

After

square=6*19+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, 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..18;exact square=144

After

square=7*19+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, 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..18;exact square=169

After

square=8*19+17;canonical square class=17;ALL root indices retained

Conditions: none

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

Step identifiers and dependencies

Rule: legendre_ALL_nonzero_square_rows_before_symbol_definition · Uses: G001, 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..18;exact square=196

After

square=10*19+6;canonical square class=6;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..18;exact square=225

After

square=11*19+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, 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..18;exact square=256

After

square=13*19+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, 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..18;exact square=289

After

square=15*19+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, 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..18;exact square=324

After

square=17*19+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, S037

S039 · implies

Before

ORIGINAL prime=19;COMPLETE nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 6], [6, 17], [7, 11], [8, 7], [9, 5], [10, 5], [11, 7], [12, 11], [13, 17], [14, 6], [15, 16], [16, 9], [17, 4], [18, 1]];nonzero square classes=[1, 4, 5, 6, 7, 9, 11, 16, 17]

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, S038

S040 · calculation

Before

Legendre ORIGINAL candidate class quotient times modulus := mul(0, 19)

After

Legendre ORIGINAL candidate class quotient times modulus = 0

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, S039

S041 · calculation

Before

Legendre ORIGINAL candidate class ORIGINAL signed value minus multiple := sub(7, 0)

After

Legendre ORIGINAL candidate class ORIGINAL signed value minus multiple = 7

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

Legendre ORIGINAL candidate class:ORIGINAL a=7,positive modulus=19;actual quotient=0

After

normalized class=7;a=quotient*modulus+class;ORIGINAL a retained in output where declared

Conditions: none

Retain the same class for negative a and normalize into0..p−1,rather than using a negative language-specific remainder.

Step identifiers and dependencies

Rule: legendre_ORIGINAL_signed_class_Euclidean_normalization · Uses: G001, S041

Reverse checks

  • 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 candidate class ORIGINAL signed value minus multiple with exact rational arithmetic, not rounded decimals.
  • S040 · premise_and_result_audit — Recompute Legendre ORIGINAL candidate class quotient times modulus with exact rational arithmetic, not rounded decimals.
  • S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

(19, 7, 7, [8, 11], 1, complete_nonzero_square_root_witness)

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: bea63131929a0aac169e04a99f1041738403ebbe267e6c420ed6e9d2bd3d8b3f
  • candidate record SHA-256: 4f8be5944adfdaf7317fcadcada22312551069646ff4f17dee4b7d371f487d4f
  • semantic review receipt SHA-256: a7c0d888ade6d307703c4222c4137c78c0d2448babdcba2fa3446fe6fdd343c1
  • question-first witness SHA-256: c9571f97a37d503ad6a61034cc00108543a05c37ef246d1c4962f55136a3de02
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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