Fermat–Euler congruences
Theorem.Fermat–Euler congruences — a^{φ(n)} ≡ 1 (mod n) for gcd(a,n)=1 · a^p ≡ a (mod p).
Proof.a^φ(n) ≡ 1 (mod n) for every a coprime to n (all n ≤ 60) and a^p ≡ a (mod p) for every prime p ≤ 60 — the foundation of modular exponentiation, exhausted within bound.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#discoveredTheoremsWaveTwentyFive
1 · Classification
finite-complete — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"Fermat–Euler congruences" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation discoveredTheoremsWaveTwentyFive.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (discoveredTheoremsWaveTwentyFive @ src/9/1) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes discoveredTheoremsWaveTwentyFive (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.