×

zbMATH — the first resource for mathematics

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.

MSC:
11B05 Density, gaps, topology
11N05 Distribution of primes
11N35 Sieves
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bombieri, E, Le grand crible dans la théorie analytique des nombres, Astérisque, 18, (1974) · Zbl 0292.10035
[2] Davenport, H, ()
[3] Erdös, P, On integers of the form 2^{k} + p and some related problems, Summa brasil math., 2, 113-123, (1950)
[4] Halberstam, H; Richert, H.-E, ()
[5] Prachar, K, Primzahlverteilung, (1957), Springer-Verlag Berlin/New York · Zbl 0080.25901
[6] Robinson, R.M, A report on primes of the form k · 2n + 1 and on factors of Fermat numbers, (), 673-681 · Zbl 0092.27505
[7] Selfridge, J.L, Solution to problem 4995, Amer. math. monthly, 70, 101, (1963)
[8] {\scJ. L. Selfridge}, private communication.
[9] Sierpiński, W, Sur un problème concernant LES nombres k · 2n + 1, Elem. math., 15, 73-74, (1960), Corrigendum 17 (1962), 85 · Zbl 0093.04602
[10] Sierpiński, W, ()
[11] Sierpiński, W, ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.