Problem 841

Let $t_n$ be minimal such that $\{n+1,\ldots,n+t_n\}$ contains a subset whose product with $n$ is a square number (and let $t_n=0$ if $n$ is itself square). Estimate $t_n$ .

Source: erdosproblems.com/841

Number theory

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An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

SOLVED, in the site's label (page last edited 14 October 2025), which describes the Statement above.

BPZ24: Bui, Hung M. and Pratt, Kyle and Zaharescu, Alexandru, A problem of Erdős-Graham-Granville-Selfridge on integral points on hyperelliptic curves. Math. Proc. Cambridge Philos. Soc. (2024), 309-323.
ErSe92: Erdős, Paul and Selfridge, J. L., Problems and Solutions: Solutions: 6655. Amer. Math. Monthly (1992), 791-794. Granville and Selfridge cite the item as P. T. Bateman, P. Erdős and J. L. Selfridge, Getting a square deal, Amer. Math. Monthly 99 (1992), 791-794.
Gu04: Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section B30 "A small set whose product is square", pp. 128--129, states the problem with $t_n$ in the page's notation, records that the Thue--Siegel theorem gives $t_n\to\infty$ faster than a power of $\ln n$ , a sentence on which the section prints comments by Granville and Silverman, and reports Selfridge's bound $t_n\le\max(P(n),3\sqrt n)$ with $P(n)$ the largest prime factor of $n$ . Library home: Guy 2004.

Problem 841

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