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**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 p_{j}-p_{i} = 2k, (p_{j} \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 (k_{n} and K_{n} 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,k_{n})/**{**n(log n)^{-2} ––>}oo and that k_{n} ––> infty. In contrast it is shown that

F(n,K_{n})/**{**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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag