Lucas theorem for binomials mod p
Theorem.Lucas theorem for binomials mod p — C(n,k) ≡ Π_i C(n_i, k_i) (mod p), where n = Σ n_i pⁱ and k = Σ k_i pⁱ are the base-p digits.
Proof.C(n,k) mod p equals the product of digit-binomials C(n_i,k_i) in base p, verified for p ∈ {2,3,5,7} and all n ≤ 40 by direct Pascal reduction vs the digit product — binomials factor through the prime base.
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
"Lucas theorem for binomials mod p" 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.