Problem 885

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claims/: The 4 claim pages of Problem 885, one per claimant's result; the problem's standing derives from them.

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Statement. 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

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

Status. 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.

Source. erdosproblems.com/885, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #885, https://www.erdosproblems.com/885.

References.

• [Br19] Bremner, Andrew, On a problem of Erdős related to common factor differences. Int. J. Number Theory (2019), 1059-1068.
• [ErRo97] Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353-359.
• [Ji99] Jiménez-Urroz, Jorge, A note on a conjecture of Erdős and Rosenfeld. J. Number Theory (1999), 140-143.

Formalization. Statement in formal-conjectures. The file proves erdos_885.variants.k_eq_4 by an explicit witness, while erdos_885, erdos_885.variants.k_eq_2 and erdos_885.variants.k_eq_3 are stated with proof sorry.

Current assessment

The site labels the problem OPEN: the answer is yes for $k\le4$ and unknown for every $k\ge5$. Erdős and Rosenfeld show that any number of integers can share two factor differences, and their paper prints two triples, found by Barry Guiduli, that share four values, which settles $k=2$ and $k=3$ (claim page). Jiménez-Urroz raises the two shared values to three for every $k$ (claim page), and Bremner gives infinitely many sets of four integers sharing four values (claim page). A Lean proof merged into formal-conjectures on 23 September 2026 settles $k=4$ again by an explicit witness (claim page); this corpus has not built it.

Three further items in the site's discussion thread settle only instances already settled by the refereed papers, so they have no claim pages. Mausberg's note of 19 April 2026, with a Lean repository produced with Aristotle, gives the $k=3$ witness $D(79200)\cap D(227205)\cap D(1258560)=\{36,468,692,1028\}$. A post by Aleksanndr_NFA of 23 September 2026, with no manuscript, gives the $k=4$ example $515504$, $3542528$, $6010004$ and $19523504$, sharing $872$, $4328$, $5003$ and $11672$. Two notes posted by zoahdev on 29 September 2026 give three integers sharing five values and do not reach $k=5$. Every membership in these examples checks by computation.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1997_factor_difference_set_integers