Problem 889

For $k\geq 0$ and $n\geq 1$ let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\leq i<k$ . Equivalently, $v(n,k)$ counts the number of prime factors of $n+k$ which are $>k$ .

Is it true that

v_0(n)=\max_{k\geq 0}v(n,k)\to \infty

as $n\to \infty$ ?

Source: erdosproblems.com/889

Number theory

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Open

No claim settles this problem.

ErSe67: Erdős, P. and Selfridge, J. L., Some problems on the prime factors of consecutive integers. Illinois J. Math. (1967), 428-430.
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 B27 "The number of prime factors of $n+k$ which don't divide $n+i$ , $0\le i<k$ ", p. 126, states the question $v_0(n)\to\infty$ and the exceptions $n=1,2,3,4,7,8,16$ to $v_0(n)>1$ . Library home: Guy 2004.

Problem 889

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Proof claims

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