The board / number theory / #437
Erdős Problem #437
PROVEDdatabase updated 2025-08-31
The statement as posed is true, and a proof exists.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The number of partial products \(a_1, a_1a_2, \ldots, a_1\cdots a_k\) of a finite sequence `a = [a₁, …, a_k]` which are squares.
`L x` is the maximal number of square partial products of a sequence \(1\leq a_1<\cdots<a_k\leq x\).
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? The answer is yes, which follows from work of Bui, Pratt, and Zaharescu [BPZ24], as noted by Tao [Ta24].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/437.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/437
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