## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  435.10025
Autor:  Erdös, Paul; Straus, E.G.
Title:  Remarks on the differences between consecutive primes. (In English)
Source:  Elem. Math. 35, 115-118 (1980).
Review:  Define F(n,k), to be the number of solution of pj-pi = 2k, (pj \leq n), and let f(n,k) be the number for which j = i+1. This paper is concerned with the behaviour as n ––> oo of the maximum values of f(n,k) and F(n,k), and with the least values (kn and Kn respectively) for which the maxima are attained. Hardy and Littlewood gave a conjectured asymptotic formula for F(n,k), for fixed k. On the assumption of this it is shown that

f(n,kn)/{n(log n)-2 ––>}oo

and that kn ––> infty. In contrast it is shown that

F(n,Kn)/{n(log log n)(log n)-2 >> 1

without any hypothesis.
Reviewer:  D.Heath-Brown
Classif.:  * 11N05 Distribution of primes
11N35 Sieves
Keywords:  differences between consecutive primes; most frequent difference; Hardy- Littlewood prime-pair conjecture

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