Euler polynomial n²+n+41 primes then breaks at 41²
Theorem.Euler polynomial n²+n+41 primes then breaks at 41² — n² + n + 41 prime for 0 ≤ n ≤ 39, and 40² + 40 + 41 = 41².
Proof.prime for every n = 0..39 and composite at n = 40 = 41² exactly — the famous long prime run with its precise computed breaking point.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#discoveredTheoremsWaveTwentyTwo
Figure — the proof, computed and plotted
f(n) = n² + n + 41
Computed by tkIsPrime @ src/9/1 — recomputed on every build, zero tokens.
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 polynomial n²+n+41 primes then breaks at 41²" 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 discoveredTheoremsWaveTwentyTwo.
- 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 (discoveredTheoremsWaveTwentyTwo @ 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 discoveredTheoremsWaveTwentyTwo (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.