Problem #853: Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime

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...

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Problem 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 $r(x)/\log x \to \infty$?

Categories: Number Theory Primes

Progress

In [Er85c] Erdős omits the condition that $t$ be even, but this is clearly necessary.

Source: erdosproblems.com/853 | Last verified: January 19, 2026

Problem Statement

Progress

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