Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 930

claims/: The 1 claim page of Problem 930, one per claimant's result; the problem's standing derives from them.

Statement. Is it true that, for every $r$ , there is a $k$ such that if $I_1,\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$ , then

\prod_{1\leq i\leq r}\prod_{m\in I_i}m

is not a perfect power?

Formulation. The intervals consist of positive integers. Erdős's display (9) on p. 28 of [Er76d] asks for no solution in positive integers, and the formal-conjectures statement requires every interval to start above $0$ . An interval containing $0$ would make the product $0$ .

Status. Open.

Source. erdosproblems.com/930, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #930, https://www.erdosproblems.com/930.

References.

[Er76d] Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.
[ErSe75] Erdős, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.

Formalization. Statement in formal-conjectures.

Current assessment

Erdős poses the question in [Er76d] as display (9), a common generalization of the theorem that a product of consecutive integers is never a power, and says that he cannot prove it even in a special case with $r=2$ . The case $r=1$ is that theorem of Erdős and Selfridge [ErSe75], with $k=2$ , recorded as the accepted partial claim Erdős and Selfridge 1975. The site notes that the lengths must be allowed to grow with $r$ : the constructions of Problem 363 show that for $r=2$ short intervals can have a perfect power as product, and that problem treats the case of squares. The question is open for every $r\geq 2$ .

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.

Linked from (7)

Problem 363 Erdős and Selfridge's theorem that a product of consecutive integers is never a power problems/diophantine_problems/E0930/claims Diophantine Problems and Powers diophantine_problems/erdos_1975_product_consecutive_integers_is_never_power Theorem 1: a product of two or more consecutive positive integers is never a power Theorem 2: a prime at least k divides the product to a power not divisible by l

Graph