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Six Primes and an Almost Prime in Four Linear Equations

Published online by Cambridge University Press:  20 November 2018

Antal Balog*
Affiliation:
Mathematical Institute Mathematics Department Budapest 1364, P. O. Box 127 Hungary, e-mail: balog@math-inst.hu Mathematics Department University of Michigan Ann Arbor, Michigan 48109-1003 U.S.A., e-mail: balog@math.Isa.umich.edu
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Abstract

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There are infinitely many triplets of primes $p,\,q,\,r$ such that the arithmetic means of any two of them, $\frac{p+q}{2},\,\frac{p+r}{2},\,\frac{q+r}{2}$ are also primes. We give an asymptotic formula for the number of such triplets up to a limit. The more involved problem of asking that in addition to the above the arithmetic mean of all three of them, $\frac{p+q+r}{3}$ is also prime seems to be out of reach. We show by combining the Hardy-Littlewood method with the sieve method that there are quite a few triplets for which six of the seven entries are primes and the last is almost prime.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1998

References

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