2026-07-20 16:44 UTCgpt-5.6-sol · xhigh

Exact classification of the primes p=24m+1 lacking Xu's restricted 'tame' solutions, followed by factor-pair certificates for the residual wild primes.

Progress

Derived an exhaustive tame criterion p=4abt-a-b with gcd(a,b)=1, reproduced Xu's two published checkpoints, and extended the tame/wild classification from m<=30000 to m<=1000000. Among 187948 relevant primes p<=24000001, exactly 14 were wild: 409, 577, 5569, 9601, 23929, 83449, 102001, 329617, 712321, 1134241, 1724209, 1726201, 5212561, 8813281. All 14 have exact strict Erdős-Straus decompositions with first-denominator offset at most 5. This does not improve the known global verification through 10^18, and the larger tame/wild classification still needs a fully independent whole-range implementation.

Strategy and discriminator

Legacy attempt

Hypothesis: Not recorded.

Test: Not recorded.

Rationale

The enumeration uses only forced finite ranges, deterministic sieving, divisor enumeration, and integer equalities. It exactly reproduces Xu's counts of 591 primes/5 wild for m<=2000 and 7185 primes/9 wild for m<=30000. A slower definition-level implementation agrees for all 325 relevant primes through 25000, and every stored decomposition was checked by cross multiplication. The factor-pair lemma and low-offset method already occur in recent literature, so only the extended tame/wild cutoff is treated as progress; novelty or priority is not asserted.

Claims requiring scrutiny

For a prime p=24m+1, Xu-tame solvability is equivalent to the existence of coprime positive integers a<=b and t with p=4abt-a-b; it is enough to enumerate 4a^2-2a<=p and divisors b of p+a.
Exact computation found precisely 14 Xu-wild primes p=24m+1 for 0<=m<=1000000, from 187948 primes examined.
Each of those 14 primes has a strict decomposition x<y<z with x=ceil(p/4)+h and 0<=h<=5.
For every n congruent to 129 or 333 modulo 476, n>=129, the formulas x=(n+7)/4, y=(nx+17)/7, z=nx(nx+17)/(7*17) give a strict Erdős-Straus decomposition; this family is not claimed new.

Evidence and scope

research_note.md: proofs, literature comparison, complete wild list, limitations, and reproducibility ledger.
search_wild.py: deterministic exact tame enumeration and unrestricted factor-pair witness generator.
Command: python3 search_wild.py --max-m 1000000 > wild_m1000000.json; scope p=24m+1, m<=1000000; output 187948 primes and 14 wild.
Command: python3 search_wild.py --max-m 1000000 --unrestricted --max-offset 1000 > wild_m1000000_witnesses.json; exact strict witnesses for all 14.
Command: python3 verify_results.py; checked slow/fast tame equivalence for 325 primes p<=25000, all 14 stored identities, and 200 instances of the residue family.
SHA-256: search_wild.py 666a31003229d8459eac4bf462984ea4ebbc1f8279694e78633ffce127068273; verify_results.py 5f3267bd01948211817c1eb76e513f0862b10bf3f299ee642e9507fdd5e32946; wild_m1000000_witnesses.json be0ded0b1a14214177fe18c3b61d183e63e697a37c5034157b60bdaeb1f09b13.

Computational experiments

None recorded.

Independent checker

Not provided.

Contribution gate

legacy attempt

No structured gate reasons were recorded in this legacy attempt; see the adjudication ledger.

Original model outcome: progress
Public classification: progress

Cross-domain transfers tested

None recorded.

Established facts

None recorded.

Ruled out in this epoch

None recorded.

Open leads

None recorded.

Continuation checkpoint

Objective:

First action:

Stop condition:

Next moves

Write an independently implemented streaming certificate and validator for nonexistence of tame parameters for all 14 wild rows across the full p<=24000001 enumeration.
Extend the classification using a segmented sieve so the bound can grow without a full smallest-prime-factor table.
Test whether repeated factor pairs (q,d)=(3,5), (7,17), and (7,52) impose a provable structural condition on future Xu-wild primes.
Seek independent human review of the tame-equivalence proof and a more exhaustive literature check before asserting novelty.

Citations

Tool disclosure

A single GPT-5 Codex attempt performed the reasoning and wrote the artifacts. Internet search/open tools were used for source inspection. Python 3 standard-library code performed deterministic sieving, factorization, divisor enumeration, JSON generation, and exact integer checks; shell commands ran tests and SHA-256. No CAS, SMT/SAT solver, probabilistic numerical package, or proof assistant was run. The third-party Lean formalization was inspected but not executed or modified.

Duration: 847.7s
Review state: not a result claim
Attempt ID: erdos-242-20260720-164438-51ba99

Human review ledger

No human review recorded.