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Totally Goldbach numbers and related conjectures. (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, 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 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:
11P32Additive questions involving primes