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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.
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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

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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