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: q1 = 3,\, q2 = 5,\, q3 = 17,...,qk is the smallest prime greater than qk-1 for which qk \not\equiv 1 (mod qi) 1 \leq i < k. Let A(x) denote the number of qi \leq x. S.W.Golomb (Zbl 067.27503) proved that liminfx ––> 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, Q1 > r+1, Q1 prime, and Qi+1 the smallest prime greater than Qi such that Qi+1 \not\equiv t (mod Qj), 1 \leq j \leq i, 1 \leq t \leq r. Let further BQ1,r(x) be the number of Q' not execceding x, then BQ1,r(x) = (1+o(1)) x / log x log2 x ··· logr+1 x where logk 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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