On the density of odd integers of the form \((p-1)2^{-n}\) and related questions. (English) Zbl 0405.10036

Given \(k\) primes \(p_1,\dots,p_k\), write \(p-1=p_1^{a_1}\dots p_k^{a_k}s_p\), where \(s_p\) is coprime to \(P=p_1p_2\dots p_k\). It is proved that the sequence of numbers occurring as \(s_p\) for some prime \(p\) has positive lower density. The most interesting unsolved problem is whether this sequence (\(s_p\)) can contain all numbers, coprime to \(P\); concerning this question some numerical data are given.


11B05 Density, gaps, topology
11N05 Distribution of primes
11N35 Sieves
Full Text: DOI


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