*(English)*Zbl 1162.11383

Goldbach’s famous conjecture is that every even integer $n$ greater than 2 is the sum of two primes; to date it has been verified for $n$ up to ${10}^{17}$; see [*T. Oliveira e Silva*, “Goldbach conjecture verification”, web page, *J. Richstein*, Math. Comput. 70, 1745–1749 (2001; Zbl 0989.11050)]. In order to establish the conjecture for a given even integer $n$, one optimistic approach is to simply choose a prime $p<n$, and check to see whether $n-p$ is prime. Of course, one has to make a sensible choice of $p$; if $n-1$ is prime, one should not choose $p=n-1$, and there is obviously no point choosing a prime $p$ which is a factor of $n$.

In this paper we examine the set of numbers $n$ for which every “sensible choice” of $p$ works:

Definition: A positive integer $n$ is totally Goldbach if for all primes $p<n-1$ with $p$ not dividing $n$, we have that $n-p$ is prime. We denote by $A$ the set of all totally Goldbach numbers.

Four conjectures are stated.

##### MSC:

11P32 | Additive questions involving primes |