Question 23701: Legendre Classes Include Zero before Nonzero Residue Classification

Original question and givens

Use bounded odd prime p=3 and the complete nonzero square table. Give (prime, class_symbol_rows, complete_partition, certificate).

• G001 — Use bounded odd prime p=3 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

source operation=legendre-classification-table;original snapshot and mathematics retained

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

Legendre ORIGINAL odd prime:ORIGINAL modulus=3;ALL possible trial divisors through square root=[]

ALL [divisor,remainder] rows=[];prime=Yes

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

Legendre ORIGINAL nonzero root1 exact square = 1

ORIGINAL root x=1 in1..2;exact square=1

square=0*3+1;canonical square class=1;ALL root indices retained

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

Legendre ORIGINAL nonzero root2 exact square = 4

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

square=1*3+1;canonical square class=1;ALL root indices retained

ORIGINAL prime=3;COMPLETE nonzero square table=[[1, 1], [2, 1]];nonzero square classes=[1]

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

Reverse checks

• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

(3, [[0, 0], [1, 1], [2, -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.