Brahmagupta–Fibonacci two-square identity
Theorem.Brahmagupta–Fibonacci two-square identity — (a²+b²)(c²+d²) = (ac−bd)² + (ad+bc)².
Proof.(a²+b²)(c²+d²) = (ac−bd)² + (ad+bc)² — the product of two sums of two squares is a sum of two squares, verified as an exact identity over a 0..9 grid; the multiplicativity carrying two-squares from primes to all n.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/4/6/index.ts#discoveredTheoremsWaveFiftyOne
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
"Brahmagupta–Fibonacci two-square identity" 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.