Taxicab(2) = 1729
Theorem.Taxicab(2) = 1729 — 1729 = 1³ + 12³ = 9³ + 10³ — the least number expressible as a sum of two positive cubes in two ways.
Proof.complete sweep — no smaller number is a sum of two positive cubes two ways (Ramanujan–Hardy computed).
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/thunder/verify/index.ts#discoveredTheoremsWaveEight
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
"Taxicab(2) = 1729" 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 discoveredTheoremsWaveEight.
- 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 (discoveredTheoremsWaveEight @ src/thunder/verify) 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 discoveredTheoremsWaveEight (src/thunder/verify/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.