Euler’s φ is multiplicative
Theorem.Euler’s φ is multiplicative — φ(mn)=φ(m)φ(n) for gcd(m,n)=1 with φ(p^k)=p^k−p^{k−1}.
Proof.φ(mn)=φ(m)φ(n) for gcd(m,n)=1 with φ(p^k)=p^k−p^{k−1} — verified over every coprime pair m,n ≤ 100 and every prime power p^k ≤ 10000, so φ is fixed by its prime-power values.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/4/6/index.ts#discoveredTheoremsWaveFiftyFour
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
"Euler’s φ is multiplicative" 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 discoveredTheoremsWaveFiftyFour.
- 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 (discoveredTheoremsWaveFiftyFour @ src/4/6) 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 discoveredTheoremsWaveFiftyFour (src/4/6/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.