Question 927: The First Positive Power Returning to One Defines the Order

Original question and givens

Use exact modulus n=4 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

• G001 — Use exact modulus n=4 and integer a=3. First require n≥2 and normalize a modulo n into the canonical nonnegative residue, then require gcd(a,n)=1. The order is the least positive return exponent, not any return. Give (power_rows, order, no_earlier_return, returns_one, certificate).

Definition and scope

Require n≥2 and a normalized residue coprime to n. The order is the first positive exponent returning to 1.

• mko-multiplicative-order-mod-n

Step-by-step derivation

n=4; a=3

order is defined here only for n≥2 and units modulo n

a=3

normalized residue=3; gcd(3,4)=1

gcd(residue,n)=1; units=[1, 3]

phi(4)=2

power product at k=1 := mul(1, 3)

power product at k=1 = 3

product=3

a^1 mod 4=3

power product at k=2 := mul(3, 3)

power product at k=2 = 9

product=9

a^2 mod 4=1

all positive k<=2: [[1, 3], [2, 1]]

first positive return to1 occurs at k=2

Reverse checks

• S008 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S007 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S006 · premise_and_result_audit — Recompute power product at k=2 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S004 · premise_and_result_audit — Recompute power product at k=1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S002 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S001 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.

Result

([[1, 3], [2, 1]], 2, Yes, Yes, order_is_the_first_positive_return)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

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.