A gcd problem of Erdős and Szekeres for binomial coefficients: every index i ≥ 10

Erdős and Szekeres asked whether, for all 1 ≤ i < j ≤ n/2, some prime p ≥ i divides gcd(C(n,i), C(n,j)) (Erdős Problem #699). We prove that the answer is yes for every i ≥ 10. For i ≥ 290 the proof combines the discriminant of a canonical integral polynomial, which has a leading coefficient dividing both binomial coefficients, with explicit prime-gap estimates. For 10 ≤ i ≤ 289 it combines three further ingredients: polynomials vanishing to high order at the lattice points {(r,s): r+s < i}, which turn the absence of a common large prime into a divisibility; effective lower bounds for the cofactors of prime powers in nearby integers, obtained from Padé approximants to binomial functions; and exhaustive interval computations for the remaining finite ranges. For 4 ≤ i ≤ 9 we prove the assertion for every n ≤ 2^1010000 and, at all heights, outside an explicit finite union of residue classes; for i = 3 we prove that a counterexample has 2^61 | n. For i ≤ 289 the proofs are computer assisted. This record contains the paper (PDF and LaTeX source) and erdos699_computations.tar.xz, split into twelve parts (join them with cat erdos699_computations.tar.xz.part* > erdos699_computations.tar.xz, MD5 1914487d90fe1dffc4d6e2d60e871d07): the programs, certificates and logs behind its computational statements, and the Lean 4 formalization of the case i ≥ 290. See README.md inside the archive. The large language models ChatGPT and Codex (OpenAI) and Claude (Anthropic) were used to develop the proofs and programs; the author takes full responsibility for their content.