Correlations of Almost Primes

Abstract

Let p 1 , . . . , p 5 be primes with p 1 p 2 , p 5 ∈ (X, 2X], let h be a non-zero integer and A > 3. Supposing that p 1 , p 3 lie in a certain range depending on X, we prove an asymptotic for the weighted number of solutions to p 3 p 4 = p 1 p 2 + h which holds for almost all 0Using the same methods and supposing instead that p 1 p 2 , p 3 p 4 have the typical factorisation we prove an asymptotic which holds for almost all 0 < |h| ≤ H with exp (log X) 1−o(1) ≤ H ≤ X log −A X and an asymptotic for the weighted number of solutions to p 3 p 4 = p 5 + h which holds for almost all 0 < |h| ≤ H with X 1/6+ε ≤ H ≤ X log −B X and B > 5.