Problem 1094

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Statement. For all $n\geq 2k$ the least prime factor of $\binom{n}{k}$ is $\leq \max(n/k,k)$, with only finitely many exceptions.

Status. Open.

Source. erdosproblems.com/1094, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1094, https://www.erdosproblems.com/1094.

References.

• [ELS88] Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., Prime factors of binomial coefficients and related problems. Acta Arith. (1988), 507-523.
• [ELS93] Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., [[../library/factorials_binomials/erdos_1993_estimates_least_prime_factor_binomial_coefficient/_index|Estimates of the least prime factor of a binomial coefficient]]. Math. Comp. (1993), 215-224.
• [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. 130: Selfridge's conjecture that $\binom nk$ with $n\ge2k$ has a prime factor $p\le n/k$ whenever $n>17.125k$, the slightly stronger conjecture that the least prime factor is at most $\max(n/k,17)$ apart from exactly four coefficients, whose least prime factors are $19$, $19$, $23$ and $29$, and the Erdős--Selfridge function $g(k)$. Library home: guy_2004_unsolved_problems_number_theory.
• [Se77] J. L. Selfridge, Some problems on the prime factors of consecutive integers. Notices Amer. Math. Soc. (1977), A456-457.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1988_prime_factors_binomial_coefficients_related_problems
• erdos_1993_estimates_least_prime_factor_binomial_coefficient
• guy_2004_unsolved_problems_number_theory