generalized pentagonal numbers g_k = k(3k−1)/2
Theorem.generalized pentagonal numbers g_k = k(3k−1)/2 — g_k = k(3k − 1)/2 for k = 1, −1, 2, −2, … — the exponents of Euler’s product.
Proof.the exponents of Euler’s product are the generalized pentagonal numbers — first eight 1,2,5,7,12,15,22,26, each pentagonal number paired with its mate: the sparse skeleton of ∏(1−x^k).
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/4/6/index.ts#discoveredTheoremsWaveFiftyThree
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
"generalized pentagonal numbers g_k = k(3k−1)/2" 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 discoveredTheoremsWaveFiftyThree.
- 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 (discoveredTheoremsWaveFiftyThree @ 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 discoveredTheoremsWaveFiftyThree (src/4/6/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.