  Erd\H{o}s considered the second moment of the gap-counting function of prime
divisors in 1946 and proved an upper bound that is not of the right order of
magnitude. We prove asymptotics for all moments.
  Furthermore, we prove a generalisation stating that the gaps between primes
$p$ for which there is no $\mathbb{Q}_p$-point on a random variety are Poisson
distributed.
