Six primes and an almost prime in four linear equations

http://dx.doi.org/10.4153/CJM-1998-025-1
Canad. J. Math. 50(1998), 465-486

  Published:1998-06-01
 Printed: Jun 1998

There are infinitely many triplets of primes $p,q,r$ such that the arithmetic means of any two of them, ${p+q\over2}$, ${p+r\over2}$, ${q+r\over2}$ 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, ${p+q+r\over3}$ 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.}