A generalization of Goldbach's conjecture

Abstract. Goldbach's conjecture states that every even number greater than \(2\) can be expressed as the sum of two primes. The aim of this paper is to propose a generalization — or a set of increasingly generalized forms — of Goldbach's conjecture and to present relevant computational results. The proposed statements also generalize Lemoine's conjecture, according to which every sufficiently large odd integer is the sum of a prime and the double of another (or the same) prime. We present computational results verifying several cases of the statement until certain values and information regarding the resulting decompositions of even and odd integers.