Symmetric and asymmetric primes
type: Item
instance of: scholarly article
title: Symmetric and asymmetric primes
zbMATH Open document ID: 0855.11044
author: P. Fletcher
author: William F. Lindgren
author: Carl Pomerance
published in: Journal of Number Theory
publication date: 1996-07-09T00:00:00Z
review text: Let \(P\) denote the set of all odd primes and define \(S: P\times P\to \mathbb{N}\) by \[ S(p, q)= \sum_{k=1}^{(p- 1)/2} [qk/ p]. \] The authors call a pair of different odd primes \((p, q)\) symmetric provided that \(S(p, q)= S(q, p)\). A prime that belongs to a symmetric pair is a symmetric prime; otherwise it is called asymmetric. The authors prove that a pair \((p, q)\), \(p\neq q\) is symmetric if and only if \(p-q\) divides \(p-1\). They also give five other characterizations of a similar type (Proposition 2.1). Numerical computations show that of the first 100,000 odd primes about 5/6 of them are symmetric. This suggests that the density of symmetric primes is positive and perhaps near 0.8. The authors prove however that this count is misleading. Theorem 3.1: For all sufficiently large numbers \(x\), the number of symmetric primes up to \(x\) is \(\leq x/(\log x)^{1.027}\). The method of the proof is based on the characterization of symmetric pairs of some sieve estimates. At the end of the paper the authors conjecture that there is some number \(\sigma> 1\) such that the number of symmetric primes up to \(x\) is \(x/ (\log x)^{\sigma+ o(1)}\) as \(x\to \infty\). They give heuristic arguments that \(\sigma= 2- (1+\log \log 2)/ \log 2= 1.08607 \dots\;\).
reviewed by: J. Kaczorowski
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11A41
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 883837
zbMATH Keywords: asymmetric primes
zbMATH Keywords: applications of sieve methods
zbMATH Keywords: density symmetric primes
MaRDI profile type: Publication
full work available at URL: jnth.1996.0066
OpenAlex ID: W2081884922
DOI: 10.1006/JNTH.1996.0066
description: scientific article; zbMATH DE number 883837