Question 995: Undefined Nonunit/Modulus Cases Differ from Order One and Negative Residues

Original question and givens

Use exact modulus n=24 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=negative_normalization. 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=24 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=negative_normalization. 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=24; a=-1

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

a=-1

normalized residue=23; gcd(23,24)=1

gcd(residue,n)=1; units=[1, 5, 7, 11, 13, 17, 19, 23]

phi(24)=8

power product at k=1 := mul(1, 23)

power product at k=1 = 23

product=23

a^1 mod 24=23

power product at k=2 := mul(23, 23)

power product at k=2 = 529

product=529

a^2 mod 24=1

all positive k<=2: [[1, 23], [2, 1]]

first positive return to1 occurs at k=2

boundary=negative_normalization; residue=23

qualified=true; order=2

Reverse checks

• 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 — 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

(negative_normalization, 24, -1, 23, 1, Yes, [[1, 23], [2, 1]], 2, 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.