Problem 369

../

claims/: The 4 claim pages of Problem 369, one per claimant's result; the problem's standing derives from them.

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Statement. 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?

Formulation. As worded, the question is trivially true: $1,\ldots,k$ are $n^\epsilon$-smooth once $n>k^{1/\epsilon}$, as the site notes. The wording follows Erdős and Graham ([ErGr80], p. 69), who ask whether for every $n\ge n_0(\epsilon)$ there are two, or more generally $k$, consecutive integers less than $n$ all of whose prime factors are less than $n^\epsilon$, and add: "The answer should be affirmative but the problem seems very hard." Those words fit only a nontrivial question, but no source says which one was meant. The site's commentary gives two nontrivial readings and does not choose between them. On the first, each member $m$ of the run is $m^\epsilon$-smooth. On the second, the run lies in $[n/2,n]$, for all large $n$. The wording and both readings are proved, so they are settled the same way and the problem counts as proved: Balog and Wooley 1998 give the first reading, and the second only for infinitely many $n$; Yang 2026 gives, for all large $n$, a run in $[n/2,n]$ each of whose members $m$ is $m^\epsilon$-smooth, which is both readings at once; and Theorem 2.1 of Bober, Fretwell, Martin and Wooley gives, for all large $n$, a run of $n^\epsilon$-smooth integers in $[n-n^c,n]$ for some $c<1$, which also gives both. The formal-conjectures statement erdos_369 encodes the second reading (see Formalization).

Status. 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 $n$.

Source. erdosproblems.com/369, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #369, https://www.erdosproblems.com/369.

References.

• [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] [[../library/arithmetic_functions/balog_1998_strings_consecutive_integers_no_large_prime_factors/_index|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: erdos_1980_old_new_problems_results_combinatorial_number_theory.

Formalization. Statement in formal-conjectures (erdos_369, category research solved, as of its 2026-09-18 commit), which formalizes the second strengthening, a run of $k$ consecutive integers in $[n/2,n]$ with every prime factor at most $n^\epsilon$ for all large $n$, and links as its formal proof the lean-proofs copy Erdos369.lean of van Doorn's file ErdosProblem369.lean. This corpus has built and audited neither file.

Current assessment

Proved in the wording and in both of the site's nontrivial readings. The Statement is the site's wording (page last edited 2026-04-28): $k$ consecutive $n^\epsilon$-smooth integers in $\{1,\dots,n\}$ for all large $n$, which $\{1,\dots,k\}$ satisfies once $n>k^{1/\epsilon}$. Erdős and Graham's words call for a nontrivial question and no source fixes one, so the problem counts as settled only when both readings under Formulation are settled the same way, and they are.

The first reading, that each member $x$ of the run be $x^\epsilon$-smooth, follows from Balog and Wooley 1998 (refereed), which gives the second only for infinitely many $n$, so it is a partial claim. Eggleton and Selfridge 1976 had given runs of five for infinitely many $n$, the accepted partial claim Eggleton and Selfridge 1976, which settles $k\le5$ of the wording and of the first reading.

The second reading, the run inside $[n/2,n]$ for all large $n$, follows from Yang's 2026 construction, accepted on the forum by the site's curator and formalized in Lean, which formal-conjectures links as the proof of its statement; each member $m$ of Yang's run is $m^\epsilon$-smooth, so it settles the first reading too. It also follows, with the run inside $[n-n^c,n]$ for some $c<1$, from Theorem 2.1 of Bober, Fretwell, Martin and Wooley 2020 (refereed), a deduction Wooley pointed out and the site's curator wrote out on the forum; that run settles the first reading as well.

No refereed publication of Yang's argument is known, and this corpus has built and audited neither Lean file. The site's discussion thread (eleven comments, 2026-03-26 to 2026-03-27) carries Yang's argument and the curator's deduction from [BFMW20]. The site relates the problem to Problems 370 and 928.

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.

• balog_1998_strings_consecutive_integers_no_large_prime_factors
• balog_1998_strings_consecutive_integers_no_large_prime_factors / lemma_2_2
• balog_1998_strings_consecutive_integers_no_large_prime_factors / theorem_1
• balog_1998_strings_consecutive_integers_no_large_prime_factors / theorem_2
• bober_2020_smooth_values_polynomials
• bober_2020_smooth_values_polynomials / theorem_2_1