# Fermat modulo 7 and a composite boundary [← Guide index](https://ocme.evemisslab.com/en/derivations/) · [Original MKO](https://ocme.evemisslab.com/en/mko/mko-fermat-little-theorem/) 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-fermat-little-theorem-01 · ocme-new142-reader-adoption-20261021 ## Goal and scope (-2)^6 is congruent to 1 modulo 7; for multiples of 7, only the congruence a^7≡a modulo 7 applies; passing a Fermat test does not prove primality. Explicit prime modulus 7, full multiplication permutation and exact squaring, with composite 341 separately. ## Givens and symbols - calc - G1 — p=7,a=-2, normalized to 5 modulo 7; the theorem statements are congruences, not integer equalities. - p — qualified prime modulus - U_7 — all nonzero residues modulo 7 ## Rules and sources - calc · prerequisite — Assume exact integer arithmetic, division with remainder, and definitions of divisibility and gcd; every congruence has an explicit modulus. Explicitly assume the complete trial-division-to-square-root primality criterion. - mko_definition · cited — The theorem modulus is an explicit prime p≥2. The coprime form requires p not dividing a; the all-integer form also handles p dividing a without canceling a. Exact modular powers normalize negative bases. Composite examples such as561 show that passing a Fermat test does not prove primality. All residue equalities here have the declared modulo-p scope. · mko-fermat-little-theorem ## Main derivation S01 · impliesBefore p=7,\quad a=-2 After 7\text{ prime},\quad a\equiv5\pmod7,\quad7\nmid a Conditions: none Rule: calc · Uses: G1 Leaf basis: calc A composite 7 would have a prime divisor at most sqrt(7); only 2 is possible and does not divide 7. The base is a unit. S02 · impliesBefore U_7=\{1,2,3,4,5,6\} After 5U_7\pmod7=(5,3,1,6,4,2) Conditions: none Rule: calc · Uses: G1, S01 Leaf basis: calc Multiplying every nonzero residue by 5 produces each exactly once, a complete permutation. S03 · impliesBefore \prod_{u\in U_7}(5u)\equiv\prod_{u\in U_7}u\pmod7 After 5^6\cdot720\equiv720\pmod7,\quad720\equiv6\pmod7 Conditions: none Rule: calc · Uses: G1, S02 Leaf basis: calc Permutation preserves the product; factoring out six copies of 5 yields the product to cancel. S04 · impliesBefore 5^6\cdot6\equiv6\pmod7 After 5^6\equiv1\pmod7\Rightarrow(-2)^6\equiv1\pmod7 Conditions: none Rule: calc · Uses: G1, S01, S03 Leaf basis: calc Multiply by 6 to cancel a unit and restore the original negative representative. S05 · impliesBefore 5^2\pmod7 After 5^2\equiv4,\quad5^4\equiv2,\quad5^6\equiv2\cdot4\equiv1\pmod7 Conditions: none Rule: calc · Uses: G1, S04 Leaf basis: calc Independent exact squaring and multiplication recheck the same result without floating-point powers. S06 · impliesBefore a=7b,\quad b\in\mathbb Z After a^7\equiv0\equiv a\pmod7,\quad a^6\equiv0\not\equiv1\pmod7 Conditions: none Rule: calc · Uses: G1, S01 Leaf basis: calc If 7 divides a, the all-integer form still holds; the coprime form does not apply and a cannot be canceled. S07 · impliesBefore n=341=11\cdot31,\quad2^{10}=1024 After 1024=3\cdot341+1\Rightarrow2^{340}=(2^{10})^{34}\equiv1\pmod{341} Conditions: none Rule: calc · Uses: G1 Leaf basis: calc Composite 341 passes the base-2 Fermat congruence, directly refuting a primality converse. S08 · impliesBefore p=7:\quad(-2)^6\equiv1,\quad a=7b:\ a^7\equiv a After \text{coprime and all-integer forms have different domains} Conditions: none Rule: mko_definition · Uses: G1, S04, S05, S06, S07 Leaf basis: mko_definition Keep the two forms and composite counterexample separate, retaining the modulus on every equality. ## Substeps and lemmas No substeps in this guide. ## Reverse audits - S08 · premise_audit · G1, S04, S05, S06, S07 — a^p=a is not an identity in Z; for example 2^7=128 is not 2. - S07 · premise_audit · G1 — 11 and 31 exceed 1, certifying compositeness; this does not refute the prime direction. - S06 · premise_audit · G1, S01 — 0^6 has a positive exponent and is zero, not a zero-division or 0^0 issue. - S05 · premise_audit · G1, S04 — Every product is reduced modulo 7; huge integer expansion is unnecessary. - S04 · premise_audit · G1, S01, S03 — Cancellation relies on an invertible product and cannot be copied to zero. - S03 · premise_audit · G1, S02 — The remainder 6 of 720 is nonzero and 6*6=1 modulo 7, hence invertible. - S02 · premise_audit · G1, S01 — All six inputs and outputs are listed, not a sample. - S01 · premise_audit · G1 — Primality and a nonzero base are qualified first, not inferred from passing Fermat. ## Error controls and negative controls Cancel units onlyIf p divides a, a cannot be canceled to obtain the coprime form. Test directionCertify primality before applying the theorem; a passing congruence does not certify primality. - 7^6=1 modulo 7. → reject — The base is zero modulo 7 and positive powers stay zero. - 2^340=1 mod 341 proves 341 prime. → reject — 341=11*31 is composite. - a^7=a as integers. → reject — For a=2,128 differs from 2; only their difference 126 is divisible by 7. ## Result and limitations (-2)^6 is congruent to 1 modulo 7; for multiples of 7, only the congruence a^7≡a modulo 7 applies; passing a Fermat test does not prove primality. ## Verification and source binding 3 declared checks for this guide; only the listed expressions are covered. - source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f - adopted source SHA-256: 902aa6dadc0e15cc6307c24fbfeae81e06264151bfb3a4a53cd657582a78760a - original source SHA-256: df79da016f65691101971d594a3f3b8b5f4afcdf8a7607582a5050bee1841e02 - 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-fermat-little-theorem-01.json)