The board / number theory / #145
Erdős Problem #145
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(s_1 < s_2 < \cdots\) be the sequence of squarefree numbers.
Let \(A(x)\) denote the set of indices \(n\) for which \(s_n \leq x\).
Let \(s_1 < s_2 < \cdots\) be the sequence of squarefree numbers. Is it true that, for any \(\alpha\geq 0\), \[ \lim_{x\to\infty} \frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha \] exists?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/145.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA005117
Original pageerdosproblems.com/145
Attempts
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