The full unit permutation modulo 10

Goal and scope

For a=-3,n=10, phi(10)=4 and a^4=1 modulo 10.

A complete reduced-residue certificate and independent modular powers for one unit base; no primality test.

Givens and symbols

• calc

• G1 — n=10,a=-3; representatives are 1 through 10 with exact gcds.

• U_{10} — complete unit representatives
• P — product of units

Rules and sources

• calc · prerequisite — Assume exact integer arithmetic, division with remainder, and definitions of divisibility and gcd; every congruence has an explicit modulus.
• mko_definition · cited — Enumerate the complete reduced residue system and check multiplication by a permutes it, or compute a^phi(n) modulo n exactly. Normalize a negative base first. Nonunits cannot use this theorem, and one passing congruence has no primality converse; Fermat is the prime-modulus special case. · mko-euler-theorem-number-theory

Main derivation

n=10,\quad a=-3

n\ge2,\quad a\equiv7\pmod{10},\quad\gcd(7,10)=1

k=1,2,3,4,5,6,7,8,9,10

\gcd(k,10)=1,2,1,2,5,2,1,2,1,10

U_{10}=\{1,3,7,9\}

\varphi(10)=4,\quad7U_{10}\pmod{10}=(7,1,9,3)

P=1\cdot3\cdot7\cdot9=189

7^4P\equiv P\pmod{10},\quad P\equiv9,\quad9^{-1}\equiv9\pmod{10}

7^4\cdot9\equiv9\pmod{10}

7^4\equiv1\pmod{10}\Rightarrow(-3)^4\equiv1\pmod{10}

7^2=49

7^2\equiv9,\quad7^4\equiv9^2\equiv1\pmod{10}

a=2,\quad\gcd(2,10)=2

2^{\varphi(10)}=16\equiv6\not\equiv1\pmod{10}

10=2\cdot5\text{ composite},\quad(-3)^4\equiv1\pmod{10}

\text{Euler congruence holds for this composite modulus}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S03, S05, S06, S07 — Fermat is the prime-modulus specialization phi(p)=p-1, not a converse primality test.
• S07 · premise_audit · G1, S03 — This does not refute Euler because the unit hypothesis fails.
• S06 · premise_audit · G1, S03, S05 — The totient comes from complete S02, not inferred backward from this power test.
• S05 · premise_audit · G1, S01, S04 — This is congruence modulo 10, not integer equality 81=1.
• S04 · premise_audit · G1, S03 — Cancellation needs an invertible product; nonunits cannot be included indiscriminately.
• S03 · premise_audit · G1, S02 — The permutation audited here is on the same set of units; this certificate does not enumerate the permutation on all ten residues.
• S02 · premise_audit · G1, S01 — Select only entries with gcd one as units.
• S01 · premise_audit · G1 — 10=7+3 and 7=2*3+1 certify gcd 1.

Error controls and negative controls

Unit premise

Check gcd(a,n)=1 before permutation or cancellation.

Scope

This theorem requires n≥2; a totient convention at n=1 does not extend its domain.

• Euler applies to every a. → reject — 2 modulo 10 is a nonunit and 2^4=6 modulo 10.
• a^4=1 proves 10 prime. → reject — 10=2*5.
• phi(10)=10. → reject — The complete units are only 1,3,7,9.

Result and limitations

For a=-3,n=10, phi(10)=4 and a^4=1 modulo 10.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 977d1dd55d45a5003a9f4880a149b3a8e21831959802a95451cfcdc9712cf56d
• original source SHA-256: e86f8dc2f3762aa9623aba948725f0429f19ae3a77d5522ab19d3138d2124453
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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