Problem 853

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Statement. Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n=t$ has no solutions for $n\leq x$.

Is it true that $r(x)\to \infty$? Or even $r(x)/\log x \to \infty$?

Status. Open.

Source. erdosproblems.com/853, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #853, https://www.erdosproblems.com/853.

References.

• [Er85c] Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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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.

• erdos_1985_my_problems_number_theory_i_would
• erdos_1985_my_problems_number_theory_i_would / question_p80_missing_gap