# The full unit permutation modulo 10 [← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-euler-theorem-number-theory/) This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented. derivation-mko-euler-theorem-number-theory-01 · ocme-new142-reader-adoption-20261021 ## 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 S01 · impliesBefore n=10,\quad a=-3 After n\ge2,\quad a\equiv7\pmod{10},\quad\gcd(7,10)=1 Conditions: none Rule: calc · Uses: G1 Leaf basis: calc Check modulus and unit hypotheses first, normalizing the negative base to 7. S02 · impliesBefore k=1,2,3,4,5,6,7,8,9,10 After \gcd(k,10)=1,2,1,2,5,2,1,2,1,10 Conditions: none Rule: calc · Uses: G1, S01 Leaf basis: calc A complete ten-entry gcd list omits neither even entries nor the endpoint. S03 · impliesBefore U_{10}=\{1,3,7,9\} After \varphi(10)=4,\quad7U_{10}\pmod{10}=(7,1,9,3) Conditions: none Rule: calc · Uses: G1, S02 Leaf basis: calc Each of the four units is hit once, so multiplication by 7 is a full permutation. S04 · impliesBefore P=1\cdot3\cdot7\cdot9=189 After 7^4P\equiv P\pmod{10},\quad P\equiv9,\quad9^{-1}\equiv9\pmod{10} Conditions: none Rule: calc · Uses: G1, S03 Leaf basis: calc Permutation preserves the product, and P's inverse 9 is verified by 9^2=1 modulo 10. S05 · impliesBefore 7^4\cdot9\equiv9\pmod{10} After 7^4\equiv1\pmod{10}\Rightarrow(-3)^4\equiv1\pmod{10} Conditions: none Rule: calc · Uses: G1, S01, S04 Leaf basis: calc Cancel P by multiplying its inverse 9, then return to original negative representative a. S06 · impliesBefore 7^2=49 After 7^2\equiv9,\quad7^4\equiv9^2\equiv1\pmod{10} Conditions: none Rule: calc · Uses: G1, S03, S05 Leaf basis: calc Independent repeated squaring checks the Euler congruence. S07 · impliesBefore a=2,\quad\gcd(2,10)=2 After 2^{\varphi(10)}=16\equiv6\not\equiv1\pmod{10} Conditions: none Rule: calc · Uses: G1, S03 Leaf basis: calc The nonunit violates the hypothesis and explicitly fails the congruence, so the formula is not unconditional. S08 · impliesBefore 10=2\cdot5\text{ composite},\quad(-3)^4\equiv1\pmod{10} After \text{Euler congruence holds for this composite modulus} Conditions: none Rule: mko_definition · Uses: G1, S03, S05, S06, S07 Leaf basis: mko_definition The positive example already uses composite 10, showing that an Euler pass does not imply primality. ## 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 premiseCheck gcd(a,n)=1 before permutation or cancellation. ScopeThis 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)](https://ocme.evemisslab.com/data/derivations/coverage-20261021/guides/derivation-mko-euler-theorem-number-theory-01.json)