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Frequencies of successive pairs of prime residues. (English) Zbl 1269.11096

Summary: We consider statistical properties of the sequence of ordered pairs obtained by taking the sequence of prime numbers and reducing modulo \(m\). Using an inclusion/exclusion argument and a cutoff of an infinite product suggested by Pólya, we obtain a heuristic formula for the “probability” that a pair of consecutive prime numbers of size approximately \(x\) will be congruent to \((a,a+d)\) modulo \(m\). We demonstrate some symmetries of our formula. We test our formula and some of its consequences against data for \(x\) in various ranges.

MSC:

11N13 Primes in congruence classes
11K45 Pseudo-random numbers; Monte Carlo methods
11N69 Distribution of integers in special residue classes

References:

[1] DOI: 10.1080/10586458.2009.10128890 · Zbl 1198.11081 · doi:10.1080/10586458.2009.10128890
[2] DOI: 10.1090/S0025-5718-1962-0148632-7 · doi:10.1090/S0025-5718-1962-0148632-7
[3] DOI: 10.2307/27641834 · Zbl 1139.11037 · doi:10.2307/27641834
[4] DOI: 10.1007/BF02403921 · JFM 48.0143.04 · doi:10.1007/BF02403921
[5] DOI: 10.2307/2308748 · Zbl 0092.04901 · doi:10.2307/2308748
[6] Rubinstein [Rubinstein and Sarnak 94] Michael, Experiment. Math. 3 pp 173– (1994)
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