On a conjecture of Pomerance
type: Item
instance of: scholarly article
title: On a conjecture of Pomerance
zbMATH Open document ID: 1311.11093
author: N. Saradha
author: Lajos Hajdu
author: R. Tijdeman
published in: Acta Arithmetica
publication date: 2012-10-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1107.5191
full work available at URL: http://hdl.handle.net/2437/150143
Mathematics Subject Classification ID: 11N13
zbMATH DE Number: 6090459
zbMATH Keywords: primes in residue classes
zbMATH Keywords: Riemann hypothesis
describes a project that uses: Maple
MaRDI profile type: Publication
OpenAlex ID: W1992669987
Wikidata QID: Q123361105
arXiv ID: 1107.5191
DOI: 10.4064/AA155-2-5
review text: The authors study the so called P-integers; recalling that (the Euler function) NEWLINE\[NEWLINE \varphi(n):=|\{ j\leq n, (j,n)=1\}| NEWLINE\]NEWLINE is the number of positive integers up to \(n\), coprime to \(n\), then we say that \(k\) is a P-integer, by definition, if the first \(\varphi(k)\) primes coprime to \(k\) form a reduced residue system modulo \(k\). \textit{C. Pomerance} proved in [J. Number Theory 12, 218--223 (1980; Zbl 0436.10020)] the finiteness of P-integers and made theNEWLINENEWLINENEWLINEConjecture. The largest P-integer is 30.NEWLINENEWLINENEWLINEUnder RH, Riemann Hypothesis, i.e., all the complex zeros of the Riemann \(\zeta\) function, defined in \(\text{Re}(s)>1\) as NEWLINE\[NEWLINE \zeta(s):=\sum_{n=1}^{\infty}n^{-s}, NEWLINE\]NEWLINE have real part equal to \(1/2\), the authors prove Pomerance's Conjecture (and even more, see Theorem 1.2). Also, relying on previous properties of P-integers proved by two of them (see quoted publications in the paper), They are able to find some numerical ranges in which they prove there are no P-integers (esp., all of them have to be less than \(10^{3500}\), see Theorem 1.1).NEWLINENEWLINEAs the authors point out, ``Pomerance's conjecture is closely related to the classical problem about the least primes in arithmetic progressions''.NEWLINENEWLINEFor the proofs, as they say ``We use the fact that the primitive residues modulo \(k\) between \(0\) and \(k\) are symmetric around \(k=2\). Our arguments are based on results about the zeros of the Riemann zeta function and estimates for the number of primes in intervals.''NEWLINENEWLINEActually, they need very precise constants in the remainders and this is performed both by a careful estimate applying well-known results and by computer aided calculations (by Maple).
reviewed by: Giovanni Coppola
description: scientific article; zbMATH DE number 6090459