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PROVED This has been solved in the affirmative.

Are there infinitely many $n$ such that, for all $k\geq 1$,\[ \omega(n+k) \ll k?\](Here $\omega(n)$ is the number of distinct prime divisors of $n$.)

#248: [Er74b][ErGr80,p.61]

number theory

Related to [69]. Erdős and Graham [ErGr80] write 'we just know too little about sieves to be able to handle such a question ("we" here means not just us but the collective wisdom (?) of our poor struggling human race).'

See also [679] and [826].

This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.

This has been resolved by Tao and Teräväinen [TaTe25], who have proved that there exists an absolute constant $C>0$ such that for infinitely many $n$, for all $k\geq 1$,\[\omega(n+k)\leq Ck.\]

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This page was last edited 28 December 2025. View history

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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:

T. F. Bloom, Erdős Problem #248, https://www.erdosproblems.com/248, accessed 2026-04-11

URL: https://www.erdosproblems.com/248

⇱ Erdős Problem #248