On the divisibility of binomial coefficients

Sílvia Casacuberta

Abstract


In Pacific J. Math. 292 (2018), 223–238, Shareshian and Woodroofe asked if for every positive integer n there exist primes p and q such that, for all integers k with 1 ≤ k ≤ n − 1, the binomial coefficient $\binom{n}{k}$ is divisible by at least one of p or q. We give conditions under which a number n has this property and discuss a variant of this problem involving more than two primes. We prove that every positive integer n has infinitely many multiples with this property.


Keywords


Binomial coefficients, divisibility, primorials.

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DOI: https://doi.org/10.26493/1855-3974.2103.e84

ISSN: 1855-3974

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