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We prove that ₍ p₍+₁-_₍ p₍ (p₍) ^{2}<, where pn denotes the nth prime. Since on average pn+1−pn is asymptotically log n, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences p−p′ between primes which includes the small gap result above.
Goldston et al. (Fri,) studied this question.