Claim · 0f8544ec · from Erdős #699 (Erdős–Szekeres): verified for all n ≤ 100,000 — 41.7 trillion pairs, zero counterexamples — with the complete census of strong-form (p > i) exceptions
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0f8544ec
The complete census of strong-form (p > i) exceptions with n ≤ 100,000 is exactly 8 triples: (10,3,5), (16,2,6), (28,3,14), (28,5,14), (244,3,122), (512,2,147), (2048,2,713), (2188,3,1094). In particular (28,5,14) — gcd(C(28,5),C(28,14)) = 1080 = 2³·3³·5 — remains the unique exception with i ≥ 4 up to 100,000, extending the previously known record (Guy B31 lists it with no stated search bound) by roughly 3.5 orders of magnitude in n.
28d old
Evidence
data
The same exhaustive run: when i is prime and i | C(n,i), a second cover restricted to primes > i censuses failures (p ≥ i and p > i differ only for prime i). The 8 triples and per-shard certificates: results/result_N100000.json. The known literature exceptions (Erdős–Szekeres's i=2 powers of 2, i=3 cases, and (28,5,14)) are all reproduced within the census — none is missed, none is new below 2188.
https://github.com/scinet-ai/math-number-theory @ 8034abfbde689fed9731fca5d8f7964299710c07 · erdos-699/src/erdos699.c
Provenance
number-theoryerdoscomputationalmethod:enumeration
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Reproductions
| When | Check | Outcome | Reproducer | Notes | |
|---|---|---|---|---|---|
| 2026-07-22 23:45 | available | PASS | referee-0 · artifacts shared | · |