Question 23377: The Complete Nonzero Square Table Has Half as Many Distinct Residues

Original question and givens

Use bounded odd prime p=5 and the declared nonzero quadratic-residue convention. Enumerate every x=1..p−1 and x^2 modulo p; normalize input classes into0..p−1. Zero is excluded from nonzero residues. Do not assume an Euler-criterion or Legendre-symbol result. Give (p, nonzero_square_rows, nonzero_residues, residue_count, half_nonzero_count, equal).

• G001 — Use bounded odd prime p=5 and the declared nonzero quadratic-residue convention. Enumerate every x=1..p−1 and x^2 modulo p; normalize input classes into0..p−1. Zero is excluded from nonzero residues. Do not assume an Euler-criterion or Legendre-symbol result. Give (p, nonzero_square_rows, nonzero_residues, residue_count, half_nonzero_count, equal).

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-quadratic-residue

Step-by-step derivation

source operation=quadratic-square-table;original snapshot and mathematics retained

explicit ordered source output fields=[p, nonzero_square_rows, nonzero_residues, residue_count, half_nonzero_count, equal]

Quadratic ORIGINAL modulus:ORIGINAL integer=5;complete possible trial divisors=[2]

ALL [divisor,remainder] rows=[[2, 1]];prime=Yes;a composite>1 has a factor at most its square root

Quadratic ORIGINAL nonzero root1 square := mul(1, 1)

Quadratic ORIGINAL nonzero root1 square = 1

ORIGINAL nonzero root x=1 in1..4;actual square=1

Euclidean square=0*5+1;nonzero square class=1;complete root-index order retained

Quadratic ORIGINAL nonzero root2 square := mul(2, 2)

Quadratic ORIGINAL nonzero root2 square = 4

ORIGINAL nonzero root x=2 in1..4;actual square=4

Euclidean square=0*5+4;nonzero square class=4;complete root-index order retained

Quadratic ORIGINAL nonzero root3 square := mul(3, 3)

Quadratic ORIGINAL nonzero root3 square = 9

ORIGINAL nonzero root x=3 in1..4;actual square=9

Euclidean square=1*5+4;nonzero square class=4;complete root-index order retained

Quadratic ORIGINAL nonzero root4 square := mul(4, 4)

Quadratic ORIGINAL nonzero root4 square = 16

ORIGINAL nonzero root x=4 in1..4;actual square=16

Euclidean square=3*5+1;nonzero square class=1;complete root-index order retained

Quadratic distinct nonzero residue count := sum(1, 1)

Quadratic distinct nonzero residue count = 2

Quadratic ORIGINAL modulus minus1 := sub(5, 1)

Quadratic ORIGINAL modulus minus1 = 4

Quadratic half of all nonzero roots := div(4, 2)

Quadratic half of all nonzero roots = 2

ORIGINAL ODD PRIME p=5;complete nonzero square table=[[1, 1], [2, 4], [3, 4], [4, 1]];distinct classes=[1, 4]

x^2=y^2 implies p divides(x-y)(x+y),so x=y or x=-y modp;odd p and nonzero x make opposite roots DISTINCT;each image has2 roots,so residue count=2=(5-1)/2=2

Reverse checks

• 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 Quadratic half of all nonzero roots with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Quadratic ORIGINAL modulus minus1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Quadratic distinct nonzero residue count 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 Quadratic ORIGINAL nonzero root4 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 Quadratic ORIGINAL nonzero root3 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 Quadratic ORIGINAL nonzero root2 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 Quadratic ORIGINAL nonzero root1 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

(5, [[1, 1], [2, 4], [3, 4], [4, 1]], [1, 4], 2, 2, Yes)

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.