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Zbl 1162.11383
van Golstein Brouwers, David; Bamberg, John; Cairns, Grant
Totally Goldbach numbers and related conjectures.
(English)
[J] Aust. Math. Soc. Gaz. 31, No. 4, 251-255 (2004). ISSN 0311-0729

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 [{\it T. Oliveira e Silva}, Goldbach conjecture verification'', web page, http://www.ieeta.pt/~tos/goldbach.html, {\it 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.
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Citations: Zbl 0989.11050

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