Large intervals between consecutive primes

Cadwell, J. H.

doi:10.1090/S0025-5718-1971-0299567-6

Large intervals between consecutive primes
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by J. H. Cadwell PDF

Math. Comp. 25 (1971), 909-913 Request permission

Abstract:

Some results in number theory, including the Prime Number Theorem, can be obtained by assuming a random distribution of prime numbers. In addition, conjectural formulae, such as Cherwell’s for the density of prime pairs $(p,p + 2)$ obtained in this way, have been found to agree well with the available evidence. Recently, primes have been determined over ranges of 150,000 numbers with starting points up to ${10^{15}}$. Statistical arguments are used to obtain a formula for the largest interval between consecutive primes in such a range, and it is found to agree well with recorded values. The same method is applied to predict the maximum interval between consecutive primes occurring below a given integer.

References

M. F. Jones, M. Lal, and W. J. Blundon, Statistics on certain large primes, Math. Comp. 21 (1967), 103–107. MR 220655, DOI 10.1090/S0025-5718-1967-0220655-3
G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), no. 1, 1–70. MR 1555183, DOI 10.1007/BF02403921
Cherwell, Note on the distribution of the intervals between prime numbers, Quart. J. Math. Oxford Ser. 17 (1946), 46–62. MR 17308, DOI 10.1093/qmath/os-17.1.46
Daniel Shanks, On maximal gaps between successive primes, Math. Comp. 18 (1964), 646–651. MR 167472, DOI 10.1090/S0025-5718-1964-0167472-8
L. J. Lander and T. R. Parkin, On first appearance of prime differences, Math. Comp. 21 (1967), 483–488. MR 230677, DOI 10.1090/S0025-5718-1967-0230677-4