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It suffices to check that each prime gap starting at p is smaller than {\displaystyle 2{\sqrt {p}}}. A table of maximal prime gaps shows that the conjecture holds to 4×1018. A counterexample near 1018 would require a prime gap fifty million times the size of the average gap. Matomäki shows that there are at most {\displaystyle x^{1/6}} exceptional primes followed by gaps larger than {\displaystyle {\sqrt {2p}}}; in particular,

{\displaystyle \sum _{\stackrel {p_{n+1}-p_{n}>x^{1/2}}{x\leq p_{n}\leq 2x}}p_{n+1}-p_{n}\ll x^{2/3}.}

 Sep 13, 2017

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