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Statement

Setting (pp. 1, 3). For a positive integer $n$, $t_n$ is the least nonnegative integer such that some subset of $\{n+1,\ldots,n+t_n\}$ has a product which, multiplied by $n$, is a perfect square; $t_n=0$ when $n$ is a square.

Conjecture 1 (p. 20). Let $c\in(0,1)$ be fixed, and let $n$ be a non-square integer sufficiently large in terms of $c$. Then

$$t_n\ge(\log n)^{1-c}.$$

The paper motivates it (pp. 19--20) by Theorem A.2 (p. 23) of its appendix: when the product $n(n+t_n)\prod_{i=1}^s(n+j_i)$ has an even number of factors ($s$ even), that theorem gives $t_n\gg\log n/\log\log n$. The authors expect the same bound for odd $s$, by analogy with the Hall-Lang conjecture for elliptic curves, and state the conjecture with a little room to spare.

Source. H. M. Bui, K. Pratt and A. Zaharescu, A problem of Erdős-Graham-Granville-Selfridge on integral points on hyperelliptic curves, Math. Proc. Cambridge Philos. Soc. 176 (2024), no. 2, 309--323; labels and pages are those of the arXiv:2211.12467v1 edition identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page image. Nothing here is independently reviewed.

Proof pointer

None: the paper states it as a conjecture.

Dependencies

None.

Bears on

• Problem 841, which asks for estimates of $t_n$: the conjecture would strengthen the lower bound of Theorem 1.4 for every large non-square $n$; it is unproved.