Problem 365

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claims/: The 2 claim pages of Problem 365, one per claimant's result; the problem's standing derives from them.

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Statement. 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?

Is the number of such $n\leq x$ bounded by $(\log x)^{O(1)}$?

Formulation. The first question is read as the site reads it: must one of $n$, $n+1$ be a square? Read loosely it would be trivially yes, since every pair $n=a^2b^3$, $n+1=c^2d^3$ solves the generalized Pell equation $d^3X^2-b^3Y^2=1$. Guy's B16 [Gu04] asks instead whether infinitely many pairs do not come from Pell equations $x^2-dy^2=\pm1$, and Walker's family answers that too. The page lists the two questions as the parts pell and count.

Status. Open, in the site's label (OPEN; page last edited 31 October 2025), which attaches to the pair of questions. The site's commentary answers the first question no, crediting Golomb's counterexample [Go70] and Walker's infinite family [Wa76]; the corpus accepts both on their refereed publication as partial claims settling the part pell, on the claim pages Golomb 1970 and Walker 1976. The second question, the $(\log x)^{O(1)}$ bound on the count, is unsettled, so the problem stays open.

Source. erdosproblems.com/365, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #365, https://www.erdosproblems.com/365.

References.

• [Go70] Golomb, S. W., Powerful numbers. Amer. Math. Monthly (1970), 848-855.
• [Gu04] Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section B16 "Powerful numbers. Squarefree numbers.", printed pp. 105--106, asks both questions of the page for Erdős's $2$-full numbers $u_i^{(2)}$, the second as whether the count of solutions with $u_i<x$ is less than $(\ln x)^c$. Library home: guy_2004_unsolved_problems_number_theory.
• [Wa76] Walker, David T., Consecutive integer pairs of powerful numbers and related Diophantine equations. Fibonacci Quart. (1976), 111-116.

Formalization. No formal-conjectures statement file exists for this problem. Collin Yuanjie Ren's Lean package formalizing Golomb's counterexample, which the community database records, is linked on Golomb's claim page; the corpus has not built it.

Current assessment

The question, as the site states it (page last edited 31 October 2025), has two parts. The first, whether one of two consecutive powerful numbers must be a square, is answered no: Golomb [Go70] observed that $12167=23^3$ and $12168=2^3\cdot3^2\cdot13^2$ are consecutive powerful numbers and neither is a square, and Walker [Wa76] showed that $7^3x^2=3^3y^2+1$ has infinitely many solutions, each giving such a pair, and described every such pair through the odd powers of a least solution of $mX^2-nY^2=\pm1$. Both papers are refereed, and the two claim pages carry the acceptance, each settling the part pell. The site's commentary also records Mahler's remark, in answer to Erdős's original question, that the Pell equation $x^2=2^3y^2+1$ already gives infinitely many consecutive powerful pairs; those pairs have a square member and bear on the count, not on the first question.

The second part, whether the number of $n\le x$ with $n$ and $n+1$ both powerful is $(\log x)^{O(1)}$, is unsettled. The Pell-equation families grow exponentially, so each contributes $O(\log x)$ pairs up to $x$, and the question is whether the pairs are confined to boundedly many such families in effect. The problem's thread (three comments as of 2026-10-07, no proof claim) records, in a comment of 2026-03-28 (post), a 2017 paper of Aktaş and Murty giving the upper bound $O(x^{2/5})$ for the count, described there as the best known; an upper bound of that shape settles no instance of the question and is not a claim. The thread's other comments concern pairs of odd powerful numbers at distance $2$ and differences of powerful numbers, which are adjacent questions.

Search scope: the site's problem page as exported (last edited 31 October 2025), its thread as of 2026-10-07, the community database entry, the formal-conjectures tree (no statement file), Ren's Lean package and the library cards for Walker and Guy; no forum proof claim and no OpenAI release item names this problem. No wider literature search was made.

Known Results

The Current assessment above records the known results.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• walker_1976_consecutive_integer_pairs_powerful_numbers_related
• walker_1976_consecutive_integer_pairs_powerful_numbers_related / example_p116
• walker_1976_consecutive_integer_pairs_powerful_numbers_related / theorem_2_2
• walker_1976_consecutive_integer_pairs_powerful_numbers_related / theorem_2_5
• walker_1976_consecutive_integer_pairs_powerful_numbers_related / theorem_3_2
• walker_1976_consecutive_integer_pairs_powerful_numbers_related / theorem_3_5
• guy_2004_unsolved_problems_number_theory