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Erd\H{o}s considered the second moment of the gap-counting function of primedivisors in 1946 and proved an upper bound that is not of the right order ofmagnitude. 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 Poissondistributed.

Gaps between prime divisors and analogues in Diophantine geometry