Problem 885

For integer $n\geq 1$ we define the factor difference set of $n$ by

D(n) = \{\lvert a-b\rvert : n=ab\}.

Is it true that, for every $k\geq 1$ , there exist integers $N_1<\cdots<N_k$ such that

\lvert \cap_i D(N_i)\rvert \geq k?

Source: erdosproblems.com/885

Number theory
Divisors

Wiki page

Status

Open

No claim settles this problem.

The site labels the problem OPEN. The instances $k=2$ and $k=3$ are settled by Erdős and Rosenfeld's two shared values and Guiduli's triples and Jiménez-Urroz's three shared values, and $k=4$ by Bremner's four integers; a Lean proof of the case k = 4 is a pending page. The general question is open.

Problem 885

Status

Proof claims

Discussion