Erdős Problem #365

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

An $r$-powerful positive integer has each nonzero prime exponent at least $r$; powerful alone means $r=2$. This does not require every exponent to be divisible by $r$, so it is weaker than being a perfect $r$th power. [S1] [S2]

Statement recorded in the supplied source.

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 $(\log x)^{O(1)}$? [S1]

Scope and interpretation.

The archive reports counterexamples to the assertion that one neighbour must be a square. Its remaining target is the polylogarithmic counting bound, not that refuted assertion. [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Improve to a polylogarithmic count or determine the true order. [S1]

Short English statement

How often can neighbouring integers both be powerful, and must these pairs arise from a square among the neighbours?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2075 (EP-365), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 365, https://www.erdosproblems.com/365. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Golomb, S. W., Powerful numbers. Amer. Math. Monthly (1970), 848-855. Bibliographic information from the supplied record.

• S4 Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. Bibliographic information from the supplied record.

• S5 Walker, David T., Consecutive integer pairs of powerful numbers and related Diophantine equations. Fibonacci Quart. (1976), 111-116. Bibliographic information from the supplied record.