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Problem 26 — Erdős Problems

Problem 26

Let $A\subset\mathbb{N}$ be infinite. Must there exist some $k\geq 1$ such that almost all integers have a divisor of the form $a+k$ for some $a\in A$ ?

Source: erdosproblems.com/26

Number theory
Divisors

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Solved·, Disproved

An accepted solution exists. The statement is false.

DISPROVED (LEAN).

DaEr51: Davenport, H. and Erdős, P., On sequences of positive integers. J. Indian Math. Soc. (N.S.) (1951), 19-24.
Er95: Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.
Te13: Tenenbaum, Gérald, Some of Erdős' unconventional problems in number theory, thirty-four years later. (2013), 651-681.

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Status: Solved, Disproved
Topics: Number theory
Divisors
In the wiki: Wiki page