Legendre three-square theorem
Theorem.Legendre three-square theorem — n is a sum of three squares iff n is not of the form 4^a(8b+7).
Proof.n is a sum of three squares iff n is not of the form 4^a(8b+7) — the brute three-square search matches the exclusion for every n ≤ 500.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/4/6/index.ts#discoveredTheoremsWaveFiftyOne
Figure — the proof, computed and plotted
n = a² + b² + c² ⟺ n ≠ 4ᵏ(8m + 7)
Computed by Legendre 4ᵏ(8m+7) test, exact arithmetic — 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
"Legendre three-square theorem" 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 discoveredTheoremsWaveFiftyOne.
- 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 (discoveredTheoremsWaveFiftyOne @ 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 discoveredTheoremsWaveFiftyOne (src/4/6/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.