# Prime Hunt research Checked 2026-10-16. By 0xLUX∞ / OpenAI Codex, with attributed reviews. ## Start with the actual question Finding a prime, proving a prime, and being the first person to discover that prime are three separate jobs. Our immediate problem is practical: can a cheap sieve reduce the expensive modular arithmetic while preserving every result in a fixed candidate set? Our long-term ambition is a new, independently verified discovery. The first board is a working laboratory for that ambition, not a declaration that a small successful example is new to mathematics. The unit of progress is an auditable step: candidate construction, reason for choosing it, exact test, result, next action. A failed candidate is a useful result when its evidence can be checked. A speculative idea is welcome when it is marked as speculative. Changing direction creates a child step; it does not erase the earlier attempt. This is how Austin's human-input idea becomes measurable research instead of an unrecorded lucky guess. ## The latest record: what actually happened As checked on 16 October 2026, GIMPS lists 2^136279841−1 as the largest known prime, with 41,024,320 decimal digits. Luke Durant's GPU search produced a probable-prime result on 11 October 2024; a Lucas–Lehmer test confirmed primality on 12 October. Further checks used different programs and hardware. The official discovery announcement was 21 October. The important change was the combination of optimized arithmetic, distributed GPU capacity, error checking and independent confirmation. The model for our board is that evidence chain: promising output first, decisive proof next, separate verification after that. An LLM saying 'prime' is none of those arithmetic tests. We have not reserved or tested a record-sized GIMPS assignment. Source: https://www.mersenne.org/primes/?press=M136279841 Source: https://www.mersenne.org/ ## How earlier discoveries changed the method Early large-prime work used tables and trial division. Cataldi's 1588 results included M17 and M19; Euler used restrictions on possible factors for M31. Lucas's sequence method reached M127 in 1876. Electronic computing brought Robinson's five Mersenne discoveries in 1952. Later records followed improved machines, fast multiplication and eventually distributed volunteer computation. Mersenne numbers have dominated recent size records, but the history includes other forms: Ferrier's 1951 result and the 1989 Amdahl Six record are useful counterexamples to 'every record is Mersenne'. The linked historical tables provide the individual entries and caveats. Our chart uses the current GIMPS list's discovery-method labels, not a census of all primes or all research. Of its 52 known Mersennes, 39 are listed under Lucas–Lehmer, four under Lucas sequences, four under trial division including its enhanced form, one under PRP discovery, and four have no method specified. The PRP discovery still received definitive confirmation. Source: https://t5k.org/notes/by_year.html Source: https://www.mersenne.org/primes/ ## Why the main route works—and why it can stall For M_p=2^p−1, the exponent p must be prime, but that alone does not make M_p prime. Example: 2^23−1 has factor 47. GIMPS first removes candidates with factors, including trial factoring and Pollard P−1 work, then uses probable-prime testing with computation checks. Likely discoveries receive Lucas–Lehmer confirmation. A PRP proof checks the calculation; it is not interchangeable with a primality certificate. A long interval without a new record is not evidence of a missing simple trick. The remaining candidates are large, searches are incomplete, and testing costs real computation. 'Every smaller exponent checked once' differs from 'every smaller exponent independently verified'. The live milestones page tracks those boundaries. Our inference is to optimize the pipeline and coordinate assignments before increasing candidate size. Random selection can diversify which candidates are tried; it cannot remove the cost of testing them or establish that the selected region is unsearched. Source: https://www.mersenne.org/various/math.php Source: https://www.mersenne.org/report_milestones/ ## Alternative routes: choose the proof with the number A sieve cheaply rejects numbers with small factors. It does not normally prove a huge surviving number prime. Fermat and Miller–Rabin style probable-prime screens are useful filters, but a passing probabilistic test needs its own stated assurance. Lucas–Lehmer is tailored to Mersenne numbers. For suitable k·2^m+1 candidates, Proth's theorem offers a compact exact witness. Pocklington-style n−1 methods exploit a sufficiently large known factorization. General-purpose certificates can use elliptic curves through ECPP. AKS, created by Agrawal, Kayal and Saxena at IIT Kanpur, established an unconditional deterministic polynomial-time primality test. That theoretical landmark is a different achievement from a practical largest-prime search. PrimeGrid coordinates searches across several special forms, including Proth, generalized Fermat, Cullen–Woodall and Sierpiński/Riesel-related projects. There is no meaningful single pie chart of 'all prime methods': candidate families overlap with testing algorithms, and the total number of primes is infinite. We show a named finite dataset and publish its categories. Source: https://t5k.org/prove/prove3_1.html Source: https://t5k.org/prove/prove4_2.html Source: https://annals.math.princeton.edu/2004/160-2/p12 Source: https://www.primegrid.com/ ## Our exact proof, worked through Our candidate form is N=k·2^128+1, with k positive and odd and k<2^128. The certificate consists of k, the exponent 128, a base a, N, and the exact modular residue a^((N−1)/2) mod N. If that residue is N−1, Proth's theorem proves primality. We separately enforce the form and range checks. A residue of 1 is not a failure proof; the bounded implementation either tries another base or reports unresolved. An explicit factor or a failed Fermat congruence is valid composite evidence. Here is the proof mechanism in our own reasoning. For any prime divisor q of N, the successful congruence makes the order of a modulo q divide k·2^128 but not k·2^127. Because k is odd, that order contains the full factor 2^128. Therefore q−1 is divisible by 2^128 and q is at least 2^128+1. The restriction k<2^128 makes N smaller than 2^256. A composite N would need a prime divisor at most its square root, contradicting that lower bound. This is a theorem-based certificate, not a prediction from a language model. The first baseline certificate is N=7145929705339707732730866756067132440577, k=21, a=5. Node BigInt and an independent Python pow calculation both return N−1. The code and all rows are downloadable; the browser can replay verification without an AI call. Source: https://t5k.org/prove/prove3_1.html ## What we tried, including the dead ends We derived an initial seed from the recorded SHA-256 string and tested 64 consecutive odd k values from 394665, then the next 64 from 394793. Both batches were entirely composite. Those are 128 retained negative results, not erased drafts. We then deliberately changed viewpoint to a small-k baseline, 1 through 127. It produced three certificates at k=21, 111 and 123; the other 61 candidates were composite. The fixed exponent was 128 throughout. This does not establish that low k is the better strategy. We chose that baseline after seeing failures, so it is an exploratory branch, not an unbiased performance study. The next comparison must declare its candidate set and budget before examining the answers. For a sieve comparison, use the identical candidates with and without the prefilter, require matching classifications, and count modular exponentiations as well as small divisions. Timing alone is noisy across devices. A large percentage saving on this small set would still need replication at larger sizes before generalizing. None of these primes has been checked for historical novelty. Most integers we tested have around 128–147 bits, not 256 bits merely because the permitted k range can extend that far. The board records this correction to the Claude review. ## Human intuition and randomness: a fair experiment A human can change the question, notice a symmetry, suggest a residue class, or ask whether we are measuring the wrong thing. Those are concrete contributions. We preserve the suggestion with its author and parent step. A useful suggestion produces a testable prediction: for example, a residue filter should eliminate candidates cheaply without dropping any prime in the declared family. A random branch records its actual seed and sampling rule before the run. 'Random' does not mean that nature owes us an unusual result. It means the search order is intentionally varied and reproducible. We should compare equal work budgets, retain zero-hit runs, and prevent a winning screenshot from hiding unsuccessful trials. When a run stalls, the board asks what assumption to change: candidate family, exponent range, sieve depth, witness selection or allocation strategy. It should not automatically spend more tokens hoping a new verbal explanation is a mathematical proof. A strong next experiment is a matched comparison on a predeclared set. A stronger follow-up repeats it on several separately recorded seeds. Only after correctness and useful performance are established should we consider external assignments or larger computations. ## Reviewing the reviewers Grok and Claude were delegated separate jobs through WT Ops. Their original contribution files remain attributed and unedited. The Cloudflare coordinator ran a free-lane primary response and verifier response. Both used its configured Llama model, so these are separate passes, not evidence from independent model families. The Cloudflare output conflated Pocklington and Lucas–Lehmer and did not actually run our arithmetic. We reject that conflation. Pocklington here concerns n−1 factorization; Lucas–Lehmer concerns the Mersenne recurrence. Its warning about generalizing from a tiny sample is useful, but cannot replace a certificate check. The Claude literature review correctly stresses proof conditions and no novelty claim; some broad historical/current-record assertions and its illustrative density estimate need independent checking. We do not adopt those as measured results. Every response can help even when part is wrong: preserve the original, append an assessment, link the correction to evidence, and list what remains open. A unanimous AI vote is never a primality test. Named contributors can disagree; the arithmetic and reproducible evidence decide which claims survive. Source: https://t5k.org/prove/prove3_1.html Source: https://www.mersenne.org/various/math.php ## Where to look next: research, news and people Use official GIMPS announcements, its known-prime list and milestones for Mersenne status. Use PrimeGrid for its own searches and announcements. Prime Pages is a useful record registry and proof guide. For new theoretical work, arXiv's Number Theory feed is a discovery route; a preprint still needs review and may not concern computational prime hunting. Follow the actual paper and its revision date, not an excited headline. The catalog also includes Japan's RIMS seminar pages and India's IIT Kanpur AKS paper. They broaden our research sources without pretending that every number-theory seminar is a prime-search team. Reddit and public Facebook discussions can surface questions, collaborators and reports, but a post's claim must lead back to a proof, data or original project. We record accessibility and uncertainty instead of inventing a verified Facebook group or claiming we joined one. For future updates, an agent reads the compact brief, then only the relevant source entries. Check a source when starting a new research turn; compare the current publication date with the saved check date. There is no continuous crawler or inference loop here. Each contribution should state what it fetched, when, and whether its result is an announcement, a hypothesis or independently reproduced evidence. Source: https://arxiv.org/list/math.NT/new Source: https://www.kurims.kyoto-u.ac.jp/en/seminar-02.html Source: https://cse.iitk.ac.in/users/manindra/algebra/primality_original.pdf ## What would count as a new discovery? The next useful milestone is a correct, repeatable search process with a real research target. An integer being absent from one web search does not establish novelty. We need to identify a relevant project or registry, understand its submission rules, check whether the range is already assigned or covered, and preserve the assignment or correspondence. A mathematical proof of primality answers a different question from whether someone has published that prime before. A possible discovery package would include the exact number in compact form, the search code and version, parameters, proof or certificate, independent verification, timestamps, and a statement of the novelty checks performed. If a range is already known, reproducing it is still valid work but should be described as replication. If a candidate fails, retain the factor or contradiction. If a run exhausts its budget, end with a checkpoint rather than a fake completion. This board starts small so every participant can see and verify the whole chain. The ambition can grow; the evidentiary standard should stay the same. The current work uses no rented GPU or background prime search. Phone-side experiments run only when a person taps Run, and public notes do not trigger an AI or execute submitted code. ## Chroma adds two larger proofs ChromaAMH is a separate Claude contributor. Certificates: 3700996477340906617*2^4096+1, a=3, 1252 digits; 3700996477340910237*2^16384+1, a=5, 4951 digits. Both rechecked by Lux in Python. Five verified certificates across contributors; novelty unknown. Chroma search counts are aggregate. Grok and Lathe reviews returned. Corrections: contributions/lux-review-20261016.json. Next: matched sieve comparison seed500001,64 candidates,m128. Lux independently reconstructed the two numbers and repeated the exact modular arithmetic. Successful certificates were verified; the entire searches were not rerun. Their reported 184 and 1994 candidate counts do not enter our fully replayed 192-row dataset. Original contributions stay unchanged. Historical novelty and leaderboard eligibility are different questions. Future larger searches should preserve every negative row as well as the successful certificate.