On linear equations with prime variables of special type (Q838455)

The present paper proves an asymptotic formula for linear equation in primes and almost primes that is in some sense a hybrid between a binary and a ternary problem. More precisely, it is proved that every large integer \(\equiv 3\;\text{or}\;5\pmod 6\) is the sum of 3 primes \(p_1, p_2, p_3\) such that \(p_1+2\) and \(p_2+2\) have at most two prime factors. Similarly, up to \(O(N(\log N)^{-A})\) exceptions, all integers \(\equiv 0\;\text{or}\;4 \pmod 6\) in the interval \([N/2, N]\) are the sum of two primes \(p_1, p_2\) such that \(p_1+2\) has at most two prime factors. This improves an earlier result of the author [Monatsh. Math. 151, No. 4, 319--332 (2007; Zbl 1188.11050)] and is also related to work of \textit{D. I. Tolev} [``Additive problems with prime numbers of special type,'' Acta Arith. 96, No. 1, 53--88 (2000); corrigendum ibid. 105, No. 2, 205 (2002; Zbl 0972.11096)]. The proof is a combination of the circle method and sieve methods.