Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T13:59:24.769Z Has data issue: false hasContentIssue false
  • English
  • Français

On the number of primes p for which p+a has a large prime factor

Published online by Cambridge University Press:  26 February 2010

Morris Goldfeld
Morris Goldfeld
Affiliation:
Columbia University, New York 27, N.Y.
Get access

Extract

For any fixed integer a and real variables x, y with y < x let Na(x, y) denote the number of primes p ≤ x for which p + a has at least one prime factor greater than y. As an elementary application of the following deep theorem of Bombieri on arithmetic progressions,

Theorem (Bombieri, [1]). For any constant A > 0, there exists a positive constant B such that ifwith l = log x, then for x > 1

where Φ(x; m, a) denotes the number of primes less than x which are congruent to a mod m;

we shall prove the following theorem:

Theorem 1. Let a be any fixed integer and let x > e. We then have

where the double sum is taken over primes p and q.

Type
Research Article
Information
Mathematika , Volume 16 , Issue 1 , June 1969 , pp. 23 - 27
Copyright
Copyright © University College London 1969

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.)

References

1.Bombieri, E., “On the large sieve”, Mathematika, 12 (1965), 201–225.CrossRefGoogle Scholar
2.Hardy, and Wright, , An introduction to the theory of numbers (Oxford, 1965), 348–349.Google Scholar
3.Pracher, K., Primzahlverteilung (Springer, 1957), 44–45.Google Scholar