An intermediate conjecture between Goldbach and Dubner: every even number is the sum of a prime and a twin prime

Abstract

We study the statement that every even number $n \ge 6$ is the sum of a prime and a member of a twin prime pair. It sits between the conjectures of Goldbach and Dubner and implies both the Goldbach and the twin prime conjectures. Our main result is conditional: if the number of twin primes up to $z$ is at least $c\,z/\log^2 z$ for all large $z$, a lower bound of the order predicted by Hardy and Littlewood, with nothing assumed about their distribution, then a positive proportion of the even numbers are so representable, with density at least an absolute multiple of $c$. The proof is Romanov's method with a Selberg sieve, and loses no factor of $\log\log$. Nothing in this direction can be unconditional, since representability on a set of positive density already implies the twin prime conjecture; we show further that a polylogarithmic bound on the least twin summand would force a power-type lower bound on the number of twin primes. We verify the statement exhaustively for all even numbers up to $10^{14}$, where the least twin summand never exceeds 23,029.