Problem 1093

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Statement. For $n\geq 2k$ we define the deficiency of $\binom{n}{k}$ as follows. If $\binom{n}{k}$ is divisible by a prime $p\leq k$ then the deficiency is undefined. Otherwise, the deficiency is the number of $0\leq i<k$ such that $n-i$ is $k$-smooth, that is, divisible only by primes $\leq k$.

Are there infinitely many binomial coefficients with deficiency $1$? Are there only finitely many with deficiency $>1$?

Status. Open: the site's label (page last edited 27 December 2025). The site's commentary credits Kevin Barreto's thread post of 16 December 2025 with a conditional answer to the second question. The post assumes two conjectures. The first strengthens the Lagarias--Soundararajan $xyz_{\mathrm{fin}}$ conjecture: for some $\kappa_0>1$ and every $\epsilon>0$, only finitely many coprime $X+Y=Z$ have every prime factor of $XYZ$ below $(\log\max(|X|,|Y|,|Z|))^{\kappa_0-\epsilon}$. The second is a lower bound $\log n\ge ck^{\alpha}$, with $\alpha>1/\kappa_0$ and all large $k$, on the least $n\ge2k$ at which $\binom nk$ has deficiency at least $2$, when such an $n$ exists. From these the post proves that only finitely many $\binom nk$ with $n\ge2k$ have deficiency at least $2$. A conditional thread post, it has no claim page.

Source. erdosproblems.com/1093, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1093, https://www.erdosproblems.com/1093.

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.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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• erdos_1988_prime_factors_binomial_coefficients_related_problems
• erdos_1993_estimates_least_prime_factor_binomial_coefficient
• guy_1991_western_number_theory_problems
• guy_1991_western_number_theory_problems / problem_86_18
• guy_1991_western_number_theory_problems / problem_91_03