Problem 369

Let $\epsilon>0$ and $k\geq 2$ . Is it true that, for all sufficiently large $n$ , there is a sequence of $k$ consecutive integers in $\{1,\ldots,n\}$ all of which are $n^\epsilon$ -smooth?

Source: erdosproblems.com/369

Number theory

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SolvedProved

An accepted solution exists. The statement is true.

Proved, in the wording and in both readings described under Formulation. The site shows PROVED (LEAN) (page last edited 2026-04-28); its (LEAN) suffix rests on a Lean proof of Yang's construction, which formal-conjectures links as the proof of its statement of the second reading. Two accepted full claims settle every reading: Yang 2026, accepted on the forum by the site's curator, and Bober, Fretwell, Martin and Wooley 2020, refereed and credited in the site's commentary. The accepted partial claims Balog and Wooley 1998 and Eggleton and Selfridge 1976 settle the wording and the first reading (Eggleton and Selfridge only for $k\le5$ ), and the second reading only for infinitely many $n$ .

BFMW20: Bober, J. W. and Fretwell, D. and Martin, G. and Wooley, T. D., Smooth values of polynomials. J. Aust. Math. Soc. (2020), 245-261.
BaWo98: Balog, Antal and Wooley, Trevor D., On strings of consecutive integers with no large prime factors. J. Austral. Math. Soc. Ser. A (1998), 266-276.
EgSe76: Eggleton, R. B. and Selfridge, J. L., Consecutive integers with no large prime factors. J. Austral. Math. Soc. Ser. A (1976), 1-11.
ErGr80: Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980). Library home: Erdős and Graham 1980.

Problem 369

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