Problem 699

Is it true that for every $1\leq i<j\leq n/2$ there exists some prime $p\geq i$ such that

p\mid \textrm{gcd}\left(\binom{n}{i}, \binom{n}{j}\right)?

Source: erdosproblems.com/699

Number theory
Binomial coefficients

Status

Open

No claim settles this problem.

Falsifiable: the site's label, which says the question is open and a single triple would refute it. The site credits GPT 5.6, prompted by Price, with the cases $j\le 3i/2$ and $n=2j$ (page edited 19 July 2026); see the Price claim page (2026). Van Doorn and Rocca's manuscripts, on their claim page (Van Doorn Rocca, 2026), reduce the problem to $i=3$ and a finite set and are not credited on the site. The standing in the frontmatter derives from the claim pages.

ErSz78: Erdős, P. and Szekeres, G., Some number theoretic problems on binomial coefficients. Austral. Math. Soc. Gaz. 5 (1978), 97-99; the conjecture is equation (4), p. 97. Library home: Erdős and Szekeres 1978.
Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section B31 "Binomial coefficients", printed p. 131: the noncoprimality of $\binom nr$ and $\binom ns$ for $0<r<s\le n/2$ and the Erdős--Szekeres question whether the greatest prime factor of the g.c.d. is always greater than $r$ , with one noticed counterexample for $r>3$ ; the identity and the counterexample are displayed formulas not reproduced here. Library home: Guy 2004.

Problem 699

Status

Proof claims

Discussion