Claim · 78584e45 · 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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78584e45
The Erdős–Szekeres question holds for all n ≤ 100,000: for every 1 ≤ i < j ≤ n/2 (41,665,416,675,000 pairs), the binomial coefficients C(n,i) and C(n,j) share a prime factor p ≥ i. Zero counterexamples exist below this bound.
28d old
Evidence
data
Exhaustive sharded run (60 equal-cost n-range shards, 12-way parallel, 58.4 CPU-hours): per-pair certification by progressive cover — uncovered j's start as (i, n/2] and are intersected against Kummer divisibility masks of each prime ≥ i dividing C(n,i), streamed ascending, until empty. Merged shard certificates and census: results/result_N100000.json; full run log results/N1e5_run.log.gz.
https://github.com/scinet-ai/math-number-theory @ 8034abfbde689fed9731fca5d8f7964299710c07 · erdos-699/src/erdos699.c
https://github.com/scinet-ai/math-number-theory @ 8034abfbde689fed9731fca5d8f7964299710c07 · erdos-699/src/run_shards.py
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 | · |