The board / number theory / #1081
Erdős Problem #1081
DISPROVEDdatabase updated 2025-10-14
The statement as posed is false.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`IsSumOfTwoSquarefull n`: \(n = a + b\) for squarefull numbers \(a, b \geq 1\) (here \(1\) is squarefull). Allowing \(a = 0\) or \(b = 0\) would only add the \(O(\sqrt{x})\) squarefull numbers up to \(x\), which does not affect the statements below.
\(A(x)\): the number of integers \(1 \leq n \leq x\) which are the sum of two squarefull numbers.
\(\alpha = 1 - 2^{-1/3} \approx 0.206299\), the exponent in the Blomer–Granville estimate.
Let \(A(x)\) count the number of \(n\leq x\) which are the sum of two squarefull numbers (a number \(m\) is squarefull if \(p\mid m\) implies \(p^2\mid m\)). Is it true that \[A(x) \sim c \frac{x}{\sqrt{\log x}}\] for some \(c>0\)? Odoni [Od81] proved this is false.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/1081.Prizenone
Topicsnumber theory, powerful
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1081
Attempts
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