Question 994: Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Original question and givens

Use exact modulus n=21 and integer a=1. 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. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

• G001 — Use exact modulus n=21 and integer a=1. 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. Mode=order_one. Give (mode, n, a, normalized_residue_or_null, gcd_or_null, unit_qualified, power_rows_or_null, order_or_undefined, boundary).

Definition and scope

For n<2 or a noncoprime residue the order is undefined; residue 1 has order 1; normalize a negative representative into [0,n) first.

• mko-multiplicative-order-mod-n

Step-by-step derivation

n=21; a=1

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

a=1

normalized residue=1; gcd(1,21)=1

gcd(residue,n)=1; units=[1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20]

phi(21)=12

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

power product at k=1 = 1

product=1

a^1 mod 21=1

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

first positive return to1 occurs at k=1

boundary=order_one; residue=1

qualified=true; order=1

Reverse checks

• 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 — Order is the first positive return to1; check n≥2 and coprimality first, and do not assert order=phi.
• 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

(order_one, 21, 1, 1, 1, Yes, [[1, 1]], 1, mode_specific_multiplicative_order_boundary)

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.