Erdős Problem #1143

Let $p_1<\cdots<p_u$ be primes and let $k\geq 1$. Let $F_k(p_1,\ldots,p_u)$ be such that every interval of $k$ positive integers contains at least $F_k(p_1,\ldots,p_u)$ multiples of at least one of the $p_i$.

Estimate $F_k(p_1,\ldots,p_u)$, particularly in the range $k=\alpha p_u$ for constant $\alpha>2$.

#1143: [Va99,1.8]

number theory | primes

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In [Va99] it is reported that Erdős and Selfridge found 'the exact bound' when $2<\alpha<3$, and that 'if $\alpha>3$ then very little is known'. No reference is given, and I cannot find a relevant paper of Erdős and Selfridge.

See also [970].

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This page was last edited 23 January 2026.

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