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 $k\geq 1$,\[\omega(n+k)\leq Ck.\]

View the LaTeX source

This page was last edited 05 December 2025.