Question 926: The First Positive Power Returning to One Defines the Order

Original question and givens

Use exact modulus n=3 and integer a=2. 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=3 and integer a=2. 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=3; a=2

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

a=2

normalized residue=2; gcd(2,3)=1

gcd(residue,n)=1; units=[1, 2]

phi(3)=2

power product at k=1 := mul(1, 2)

power product at k=1 = 2

product=2

a^1 mod 3=2

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

power product at k=2 = 4

product=4

a^2 mod 3=1

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