Goldbach’s famous conjecture is that every even integer greater than 2 is the sum of two primes; to date it has been verified for up to ; see [T. Oliveira e Silva, “Goldbach conjecture verification”, web page, http://www.ieeta.pt/~tos/goldbach.html, J. Richstein, Math. Comput. 70, 1745–1749 (2001; Zbl 0989.11050)]. In order to establish the conjecture for a given even integer , one optimistic approach is to simply choose a prime , and check to see whether is prime. Of course, one has to make a sensible choice of ; if is prime, one should not choose , and there is obviously no point choosing a prime which is a factor of .
In this paper we examine the set of numbers for which every “sensible choice” of works:
Definition: A positive integer is totally Goldbach if for all primes with not dividing , we have that is prime. We denote by the set of all totally Goldbach numbers.
Four conjectures are stated.