Leave room for a discovery.

Abstract

We describe an open volunteer search for integer solutions to x³ + y³ + z³ = 114. The implementation combines a cubic-field norm construction, exact arithmetic, modular rejection, bounded client tasks, durable result receipts and independent replay. Scheduling adapts to measured execution cost while preserving exploration. The selected domains are finite and do not constitute an exhaustive height search. No solution is claimed in this draft.

1. Problem and motivation

The sums-of-three-cubes problem asks for integer triples representing a prescribed integer. The congruence obstruction modulo 9 excludes values congruent to ±4; 114 passes this necessary test. Its passing the test does not prove the existence of an integer solution. This project makes a previously local computational experiment reproducible and open to volunteer contributions.

2. Search construction

For S = x + y and V = x − y, the defining identity is

Writing D = |S| reduces z to roots r of r³ ≡ 114 (mod D), with z = r + Dq. To generate selected roots, let α³ = 114 and use

Fixed class lattices and finite coefficient domains define each task. Congruence filters precede exact square, integrality and parity checks. Every emitted triple must satisfy the original integer identity. The mathematical domain and implementation limits accompany the source.

3. Volunteer execution and verification

Opt-in browser clients use BigInt arithmetic in a worker; the portable local runner uses Python integers. Bounded deterministic task receipts are banked as GitHub issues. The authenticated issue author supplies attribution; posting establishes submission, not correctness. A scheduled verifier independently replays accepted tasks, deduplicates task identifiers and preserves exact positive identities. Display names are self-reported; leaderboard identity comes from the issue author.

4. Adaptive allocation

The cluster publishes a versioned policy after each epoch of 64 newly verified unique tasks. At least 40% of allocation is reserved for exploration. Observed replay costs inform the remaining allocation. This objective is a throughput proxy under explicit assumptions, not an estimate of the probability of discovering a solution. A separate exploratory discovery-learning experiment failed its predeclared acceptance criterion.

5. Validation and limitations

The preceding native campaign recovered all 662 known-positive fixtures in its regression corpus and included arithmetic-boundary, differential and crash-recovery checks. Those results apply to the archived native release; the portable public client has its own independent cross-language tests. Neither set is a formal proof of the complete software stack. Public receipts only certify the finite task actually replayed. Unknown global coverage and unknown success probability remain central limitations.

7. Reproducibility and attribution

Code, fixed task definitions, verification tests, strategy revisions, research notes and their failed hypotheses are maintained at github.com/Kuberwastaken/math-gambling. Final authorship and discovery attribution require evidence and agreement; participation alone does not confer authorship.