PROVED
This has been solved in the affirmative.
Let $1\leq a_1<\cdots<a_k\leq x$. How many of the partial products $a_1,a_1a_2,\ldots,a_1\cdots a_k$ can be squares? Is it true that, for any $\epsilon>0$, there can be more than $x^{1-\epsilon}$ squares?
Erdős and Graham write it is 'trivial' that there are $o(x)$ many such squares, although this is not quite trivial, using Siegel's theorem.
A positive answer follows from work of Bui, Pratt, and Zaharescu
[BPZ24], as noted by Tao in this
blog post. In particular Tao shows that, if $L(x)$ is the maximal number of such squares possible, and $u(x)=(\log x\log\log x)^{1/2}$, then\[x\exp(-(2^{1/2}+o(1))u(x)) \leq L(x) \leq x\exp(-(2^{-1/2}+o(1))u(x)).\]See also
[841].
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #437, https://www.erdosproblems.com/437, accessed 2026-04-11
