An Explicit Version of Chen’s Theorem

Abstract

We focus on Chen's theorem, proved in 1966 by Chen [5,6]. We obtain the first completely explicit version of Chen's theorem and, in doing so, improve many mathematical tools that are needed for the task. We prove the following result. THEOREM 1. All even numbers bigger than exp(36) can be written as the sum of a prime and another integer that is the product of at most two primes.

Some explicit results on the sum of a prime and an almost prime

“…Here and throughout, p denotes a prime number, 0 < δ < 2, α > 0 and X 2 are parameters we choose later, N ≥ X 2 is even, As on page 18 of [BJS22], we let k 0 := k 0 (N ) be the exceptional modulus up to…”

“…The following list of definitions is adapted directly from [BJS22]. However, we have made small modifications so that everything is expressed in terms of α 1 , α 2 , Y 0 and C(α 1 , α 2 , Y 0 ) rather than the special case α 1 = 10, α 2 = 8, Y 0 = 10.4, C = 3.2 • 10 −8 used in [BJS22]. As in [BJS22] we also take β 0 to be bounded by…”

“…However, we have made small modifications so that everything is expressed in terms of α 1 , α 2 , Y 0 and C(α 1 , α 2 , Y 0 ) rather than the special case α 1 = 10, α 2 = 8, Y 0 = 10.4, C = 3.2 • 10 −8 used in [BJS22]. As in [BJS22] we also take β 0 to be bounded by…”

“…Despite this, in [BJS22], Bordignon and the authors of this paper recently built upon unpublished work of Yamada [Yam15] to prove an effective and explicit variant of Chen's Theorem. Namely, they showed [BJS22, Corollary 4] that Chen's Theorem holds for all even N ≥ exp(exp(34.5)).…”

“…In this paper, by using a more sophisticated procedure that essentially generalises the work in [BJS22], we improve on this result as follows.…”