Question 936: The First Positive Power Returning to One Defines the Order

Original question and givens

Use exact modulus n=13 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=13 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=13; a=2

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

a=2

normalized residue=2; gcd(2,13)=1

gcd(residue,n)=1; units=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

phi(13)=12

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

power product at k=1 = 2

product=2

a^1 mod 13=2

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

power product at k=2 = 4

product=4

a^2 mod 13=4

power product at k=3 := mul(4, 2)

power product at k=3 = 8

product=8

a^3 mod 13=8

power product at k=4 := mul(8, 2)

power product at k=4 = 16

product=16

a^4 mod 13=3

power product at k=5 := mul(3, 2)

power product at k=5 = 6

product=6

a^5 mod 13=6

power product at k=6 := mul(6, 2)

power product at k=6 = 12

product=12

a^6 mod 13=12

power product at k=7 := mul(12, 2)

power product at k=7 = 24

product=24

a^7 mod 13=11

power product at k=8 := mul(11, 2)

power product at k=8 = 22

product=22

a^8 mod 13=9

power product at k=9 := mul(9, 2)

power product at k=9 = 18

product=18

a^9 mod 13=5

power product at k=10 := mul(5, 2)

power product at k=10 = 10

product=10

a^10 mod 13=10

power product at k=11 := mul(10, 2)

power product at k=11 = 20

product=20

a^11 mod 13=7

power product at k=12 := mul(7, 2)

power product at k=12 = 14

product=14

a^12 mod 13=1

all positive k<=12: [[1, 2], [2, 4], [3, 8], [4, 3], [5, 6], [6, 12], [7, 11], [8, 9], [9, 5], [10, 10], [11, 7], [12, 1]]

first positive return to1 occurs at k=12

Reverse checks

• S028 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S027 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S026 · premise_and_result_audit — Recompute power product at k=12 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S024 · premise_and_result_audit — Recompute power product at k=11 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S022 · premise_and_result_audit — Recompute power product at k=10 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S020 · premise_and_result_audit — Recompute power product at k=9 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S018 · premise_and_result_audit — Recompute power product at k=8 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S016 · premise_and_result_audit — Recompute power product at k=7 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S014 · premise_and_result_audit — Recompute power product at k=6 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S012 · premise_and_result_audit — Recompute power product at k=5 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S010 · premise_and_result_audit — Recompute power product at k=4 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• S008 · premise_and_result_audit — Recompute power product at k=3 with exact rational arithmetic, not rounded decimals.
• 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, 4], [3, 8], [4, 3], [5, 6], [6, 12], [7, 11], [8, 9], [9, 5], [10, 10], [11, 7], [12, 1]], 12, 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.