Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 376

Statement. Are there infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $105$ ?

Status. Open.

Source. erdosproblems.com/376, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #376, https://www.erdosproblems.com/376.

References.

[BeHa98] Berend, Daniel and Harmse, Jørgen E., On some arithmetical properties of middle binomial coefficients. Acta Arith. 84 (1998), no. 1, 31-41, doi:10.4064/aa-84-1-31-41.
[BlCr25] T. F. Bloom and E. Croot, Integers with small digits in multiple bases. arXiv:2509.02835 (2025).
[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\sp{2n}\sb{n})$ . Math. Comp. (1975), 83-92.
[Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Section B33 "Largest divisor of a binomial coefficient", printed p. 135, reports Graham's offer, the base 3, 5 and 7 digit conditions and the 14 known values below $7^{10}$ . The amount of the offer is in conflict: Guy prints an amount different from the one the site attributes to [Gu04] and [BeHa98]; Berend and Harmse (Section 4, from a personal communication of Graham) and Pomerance [Po15c] (Section 4) both print the site's amount. Library home: guy_2004_unsolved_problems_number_theory.
[Po15c] Pomerance, Carl, Divisors of the middle binomial coefficient. Amer. Math. Monthly (2015), 636-644.

Formalization. Statement in formal-conjectures.

Progress

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