Journal of Integer Sequences, Vol. 16 (2013), Article 13.5.2

Maximal Gaps Between Prime k-Tuples: A Statistical Approach

Alexei Kourbatov
15127 NE 24th St #578
Redmond, WA 98052


Combining the Hardy-Littlewood k-tuple conjecture with a heuristic application of extreme value statistics, we propose a family of estimator formulas for predicting maximal gaps between prime k-tuples. Extensive computations show that the estimator a log(x/a) − ba satisfactorily predicts the maximal gaps below x, in most cases within an error of ±2a, where a = Ck logkx is the expected average gap between the same type of k-tuples. Heuristics suggest that maximal gaps between prime k-tuples near x are asymptotically equal to a log(x/a), and thus have the order O(logk+1x). The distribution of maximal gaps around the “trend” curve a log(x/a) is close to the Gumbel distribution. We explore two implications of this model of gaps: record gaps between primes and Legendre-type conjectures for prime k-tuples.

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(Concerned with sequences A005250 A091592 A113274 A113404 A192870 A200503 A201051 A201062 A201073 A201251 A201596 A201598 A202281 A202361.)

Received January 22 2013; revised version received May 1 2013. Published in Journal of Integer Sequences, May 9 2013.

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