Legendre prime-power formula for n!
Theorem.Legendre prime-power formula for n! — v_p(n!) = Σ_{k≥1} ⌊n/p^k⌋.
Proof.the exponent of a prime p in n! equals Σ⌊n/p^k⌋, matched against direct factorization for all n ≤ 60 (exp of 2 in 10! is 8).
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#discoveredTheoremsWaveThirtySix
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
"Legendre prime-power formula for n!" 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 discoveredTheoremsWaveThirtySix.
- 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 (discoveredTheoremsWaveThirtySix @ 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 discoveredTheoremsWaveThirtySix (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.