Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-02T21:16:39.658Z Has data issue: false hasContentIssue false
  • English
  • Français

STRINGS OF CONGRUENT PRIMES

Published online by Cambridge University Press:  01 April 2000

D. K. L. SHIU
D. K. L. SHIU
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA
Get access

Abstract

In 1920 Chowla made the following conjecture. Let pn denote the nth prime; if q [ges ] 3, (q, a) = 1 then there are infinitely many pairs of consecutive primes pn and pn+1 such that

formula here

By considering the sum

formula here

where χ is the non-principal character modulo 4 or 6, it is possible to prove the conjecture for q = 4 and q = 6 (a = ±1). In this paper we prove Chowla's conjecture for all q and a with (q, a) = 1. Moreover, we shall show that for any k there exist ‘strings’ of congruent primes such that

formula here

For each modulus q the method used applies best to the following two sets of residue classes:

formula here

Larger values of k in terms of pn+1 can be found for residue classes belonging to these sets.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 61 , Issue 2 , April 2000 , pp. 359 - 373
Copyright
The London Mathematical Society 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)