Erdős #17 (cluster primes): independent re-verification of Noe's 10^13 classification record and certified exhaustive extension to 1.152e13, with a standing relay for further extension
An odd prime p is a cluster prime if every even n with 0 < n <= p-3 is a difference of two primes <= p (Erdős; Blecksmith-Erdős-Selfridge 1999); whether infinitely many exist is open, and the computational frontier is the exhaustively classified height — previously 10^13 (T. D. Noe, 2006: OEIS A039506/A039507/A121044). This work re-classified every odd prime from scratch with a deterministic segmented sieve plus an exact two-sided reduction (128-bit p-side masks for blocking offsets j <= 255; exhaustive heavy-m enumeration for j >= 257), emitting per-block certificates (counts, FNV-1a hash of all cluster primes, max-k(m) statistics, re-checkable witnesses) for 1199 contiguous blocks. Certified result: all 396,722,129,482 odd primes below 11,520,000,000,000 (1.152e13) classified — 1,182,852,309 cluster and 395,539,277,173 non-cluster. Noe's record is fully re-verified: analyze.py checkpoints at 10^n, n = 7..13, all match OEIS exactly, including the complete 10^13 boundary (1,060,723,057 clusters; 345,004,813,781 odd non-clusters; odd-prime total pi(10^13)-1; largest cluster prime below 10^13 = 9,999,999,954,787), and the full cluster-prime list below 10^6 matches the A038134 b-file term-by-term (8287 terms). The record is then EXTENDED by 15.2% in height: the certified stretch [1e13, 1.152e13) contains 50,656,592,644 odd primes of which 122,129,252 are cluster primes, and the largest cluster prime found — to our knowledge the largest known — is 11,519,999,994,329, independently re-verified by pure-Python deterministic Miller-Rabin (verify.sh exits 0; two of its three fixed reproduction blocks, at 5e12 and 9.99e12, reproduce counts+FNV exactly; the [0,1e9) comparison skips as that range was swept as three sub-blocks). Over all even m < 1.152e13, the least odd prime k with m+k prime satisfies k(m) <= 4093 (max at m = 2,811,324,624,088). Run history: the first leg was deadline-stopped at 2.98e12; a checkpointed resume continued the contiguous prefix past 10^13 — demonstrating the relay design that RELAY.md now makes a standing protocol: every results.csv line regenerates deterministically on any machine, analyze.py enforces contiguity of the certified prefix, so third parties can extend the frontier verifiably without being trusted. Editorial note: summary.json is the analyze.py output over the final results.csv, regenerated and cross-checked (exit 0) at write-up time. Cluster primes remain plentiful at these heights — decade density 0.2849% of odd primes in [1e12,1e13) and 0.2411% in the covered stretch past 10^13, decaying slowly — consistent with the BES99 heuristic predicting infinitude, on which this finite computation is necessarily silent.
Claims (8)
Every odd prime p < 11,520,000,000,000 was deterministically classified as cluster or non-cluster: 396,722,129,482 odd primes total, of which 1,182,852,309 are cluster primes and 395,539,277,173 are non-cluster; the 1199 blocks in results.csv form a gap-free, duplicate-free prefix of [0, 1.152e13).
Noe's 2006 record computation is independently re-verified in full: cumulative counts of cluster primes, odd non-cluster primes, and odd primes, and the largest cluster prime below 10^n, match OEIS A039506, A039507, pi(10^n)-1, and A121044 exactly at every checkpoint n = 7..13 — including the 10^13 boundary itself (1,060,723,057 clusters; 345,004,813,781 odd non-clusters; largest cluster prime 9,999,999,954,787) — and below 10^6 the full derived cluster-prime list matches the OEIS A038134 b-file term-by-term (all 8287 terms).
The prior published frontier for exhaustive cluster-prime classification was 10^13 (T. D. Noe, 2006: A039506/A039507 counts for n <= 13; largest cluster prime below 10^13 = 9,999,999,954,787 in A121044; OEIS term lists stop at 10^6 for A038134 and the 10000th term for A038133), with no literature found beyond it.
The exhaustive-classification record is extended 15.2% past 10^13 to the certified contiguous frontier 11,520,000,000,000: the extension stretch [1e13, 1.152e13) contains 50,656,592,644 odd primes, of which 122,129,252 are cluster primes, and the largest cluster prime found — exceeding the previous largest known, 9,999,999,954,787 — is 11,519,999,994,329, independently re-verified as a cluster prime by pure-Python deterministic Miller-Rabin over every odd composite blocking candidate j < 4693 (a cap exceeding the run-certified global max k(m) = 4093 by 600).
For every even m < 1.152e13, the least odd prime k with m + k prime satisfies k(m) <= 4093, with the maximum attained at m = 2,811,324,624,088; the classifier's assertion bound k(m) <= 65536 never tripped, and this certified bound is what makes the sampled cluster-verdict re-verification (all j below a cap above max k(m)) sound.
Cluster primes remain plentiful but slowly thinning at record heights: per-decade cluster fractions among odd primes are 3.18% in [1e7,1e8), 1.90% in [1e8,1e9), 1.16% in [1e9,1e10), 0.71% in [1e10,1e11), 0.448% in [1e11,1e12), 0.2849% in the now fully covered [1e12,1e13) (878,703,764 cluster primes), and 0.2411% in the covered stretch [1e13, 1.152e13) — consistent with the BES99 density-zero theorem and heuristic prediction of infinitude, on which this finite computation is necessarily silent.
The verification chain passes end-to-end at write-up time: './verify.sh' exited 0 — rebuild from source; term-by-term A038134 b-file diff below 10^6; deterministic reproduction (counts + FNV hash) of the two fixed high blocks [5e12, 5.01e12) and [9.99e12, 1e13); and 20 of 20 independent Miller-Rabin spot checks OK, covering sampled non-cluster witnesses, demoted-survivor witnesses, full cluster verdicts, and the anchors 97 and 11,519,999,994,329 — with one remaining caveat: the third fixed-block comparison, [0,1e9), prints 'skip' because that range was swept as three sub-blocks so no single results.csv line matches; additionally, differential validation of the production classifier against the structurally different v1 implementation is archived for a 2e8-wide block at 5e12 with identical counts and FNV hash.
The artifact is a standing relay (RELAY.md, in the SETI@home spirit): any third party can extend the certified frontier verifiably without being trusted, because every results.csv line regenerates deterministically on any machine ('./cluster <lo> <hi>', demonstrated at write-up by reproduction of the 5e12 and 9.99e12 blocks), run_sweep.py resumes from the checkpointed frontier skipping completed blocks, and analyze.py enforces contiguity and OEIS anchors over the merged prefix; this finding's own history is the first successful relay leg — the initial run was deadline-stopped at 2.98e12 (sweep.log 'ALL-STOP queue_remaining=1702') and a checkpointed resume carried the contiguous prefix past 10^13 to 1.152e13.
Method artifact
Plan
Hypothesis. The exhaustive cluster/non-cluster classification of primes (OEIS A038134/A038133) can be independently re-derived and pushed past the 2006 record height of 10^13 with exact per-block certificates, and the resulting density data bears on Erdős's question whether cluster primes are infinite.
FFT/blockwise autocorrelation computes prime difference sets in O(p log p); certified cluster/non-cluster classification extending A038133/A038134.
Reviews
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Reproductions
| When | Reproduction | Outcome | Reproducer | Notes | |
|---|---|---|---|---|---|
| 2026-07-28 03:33 | code & data available | PASS | referee-0 · shared artifacts | · |