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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 100.27104

**Autor: ** Erdös, Pál

**Title: ** On a problem of S.W.Golomb. (In English)

**Source: ** J. Aust. Math. Soc. 2, 1-8 (1961).

**Review: ** A set of primes is defined in the following way: q_{1} = 3,\, q_{2} = 5,\, q_{3} = 17,...,q_{k} is the smallest prime greater than q_{k-1} for which q_{k} \not\equiv 1 (mod q_{i}) 1 \leq i < k. Let A(x) denote the number of q_{i} \leq x. *S.W.Golomb* (Zbl 067.27503) proved that **liminf**_{x ––> oo} {A(x) log x \over x} = 0.

In this paper the author proves that A(x) = (1+o(1)) {x \over log x log log x}. The proof is based on use of Brun's method and results on primes in short arithmetic progression.

In the end the author states that by similar arguments the following more general result can be proved: Let r > 1, Q_{1} > r+1, Q_{1} prime, and Q_{i+1} the smallest prime greater than Q_{i} such that Q_{i+1} \not\equiv t (mod Q_{j}), 1 \leq j \leq i, 1 \leq t \leq r. Let further B_{Q1,r}(x) be the number of Q' not execceding x, then B_{Q1,r}(x) = (1+o(1)) x / log x log_{2} x ··· log_{r+1} x where log_{k} x denotes the k time iterated logarithm.

There are several misprints in the paper.

**Reviewer: ** S.Selberg

**Classif.: ** * 11N56 Rate of growth of arithmetic functions

**Index Words: ** number theory

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