The first power returning to one

Goal and scope

ord_8(-5)=ord_8(3)=2, with phi(8)=4 and 2 dividing 4 but not equal to it.

Complete first-return and unit enumeration at bounded modulus 8, with undefined and order-one boundaries.

Givens and symbols

• calc

• G1 — n=8,a=-5, using multiplicative order as the least positive return to 1.

• m — least positive return exponent
• U_8 — complete units

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• calc · prerequisite — Assume exact integer arithmetic, division with remainder, and definitions of divisibility and gcd; every congruence has an explicit modulus.
• mko_definition · cited — Require n≥2, normalize a modulo n and check coprimality first. Starting at exponent1 build every power through its first return to 1; an arbitrary later return is not the order. Count phi(n) by complete residue enumeration and verify the order divides phi(n). Noncoprime residues and n<2 have undefined order here. Residue1 has order1, and a negative integer representative is normalized first. Do not infer order=phi(n), a primitive root or additive order. · mko-multiplicative-order-mod-n

Main derivation

n=8,\quad a=-5

a\equiv3\pmod8,\quad\gcd(3,8)=1,\quad n\ge2

k=1

3^1\equiv3\not\equiv1\pmod8

k=2

3^2=9\equiv1\pmod8

3^1\ne1,\quad3^2=1\quad(\bmod8)

\operatorname{ord}_8(3)=2=\operatorname{ord}_8(-5)

k=1,2,3,4,5,6,7,8

\gcd(k,8)=1,2,1,4,1,2,1,8;\quad U_8=\{1,3,5,7\},\ \varphi(8)=4

m=\operatorname{ord}_n(a),\quad a^k\equiv1,\quad k=qm+r,\ 0\le r<m

1\equiv a^k\equiv(a^m)^qa^r\equiv a^r\Rightarrow r=0\Rightarrow m\mid k

m=2,\quad\varphi(8)=4

2\mid4,\quad2<4,\quad3^4\equiv1\pmod8

a\equiv1\pmod8;\quad a=2\text{ or }n=1

\operatorname{ord}_8(1)=1;\quad\operatorname{ord}_8(2),\ \operatorname{ord}_1(a)\text{ undefined here}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S01, S04, S07 — Undefined is not order zero, nor is it additive order modulo 8.
• S07 · premise_audit · G1, S03, S05, S06 — This example makes no automatic primitive-root or maximal-order claim.
• S06 · premise_audit · G1, S04, S05 — q is nonnegative; r>0 would contradict first-return minimality.
• S05 · premise_audit · G1, S01 — 4 counts units and need not equal this base's least return.
• S04 · premise_audit · G1, S01, S02, S03 — The negative representative shares class 3, and order depends on the class.
• S03 · premise_audit · G1, S02 — The complete smaller-positive-exponent set is {1}, with no omission.
• S02 · premise_audit · G1, S01 — a^0=1 cannot be misreported as order zero.
• S01 · premise_audit · G1 — A nonunit does not start this order search.

Error controls and negative controls

First return

Start at exponent 1 and retain all powers through the first 1.

Multiplicative not additive

3 has additive order 8 modulo 8 but multiplicative order 2; keep operations distinct.

• ord_8(3)=4 because 3^4=1. → reject — 3^2 already returns, so 4 is not least.
• Every unit order equals phi(n). → reject — Here 2<4.
• ord_8(2)=0. → reject — 2 is not a unit, so this order is undefined here.

Result and limitations

ord_8(-5)=ord_8(3)=2, with phi(8)=4 and 2 dividing 4 but not equal to it.

Verification and source binding

3 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: ce541cbfb8a5edcfd8f8ac31cb69007abb271f7722aee1a86a2a3de8d979582d
• original source SHA-256: a7c2b4919c706f641010eca03abbb0ccf5ca4a241cd70ff96070b6ead7244523
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)