The board / number theory / #365
Erdős Problem #365
OPENdatabase updated 2025-09-20
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Do all pairs of consecutive powerful numbers \(n\) and \(n+1\) come from solutions to Pell equations? In other words, must either \(n\) or \(n+1\) be a square? The answer to the first question is no: Golomb [Go70] observed that both \(12167=23^3\) and \(12168=2^3 3^2 13^2\) are powerful. We require \(n>0\) since `Nat.Powerful` also includes \(0\).
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/365.Prizenone
Topicsnumber theory, powerful
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/365
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\(S(10^{16})=26\) is exhaustively certified (five pairs have neither endpoint square), the best located theorem is \(S(x)\ll_\epsilon x^{29/100+\epsilon}\), and the unresolved step is a uniform polylo
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