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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The distribution of small gaps between successive primes

Author: Richard P. Brent
Journal: Math. Comp. 28 (1974), 315-324
MSC: Primary 10-04; Secondary 10A25, 10H15
MathSciNet review: 0330017
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Abstract: For $ r \geqq 1$ and large N, a well-known conjecture of Hardy and Littlewood implies that the number of primes $ p \leqq N$ such that $ p + 2r$ is the least prime greater than p is asymptotic to

$\displaystyle \int_2^N {\left( {\sum\limits_{k = 1}^r {\frac{{{A_{r,k}}}}{{{{(\log x)}^{k + 1}}}}} } \right)} \;dx,$

where the $ {A_{r,k}}$ are certain constants. We describe a method for computing these constants. Related constants are given to 10D for $ r = 1(1)40$, and some empirical evidence supporting the conjecture is mentioned.

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Keywords: Prime, distribution of primes, Hardy-Littlewood conjecture, prime gap, twin primes
Article copyright: © Copyright 1974 American Mathematical Society