Consequences of Goldbach's Conjecture In a letter of 1742 to Euler, Goldbach expressed the belief that ‘Every integer $N>5$ is the sum of three primes’. Euler replied that this is easily seen to be equivalent to the following statement : ‘Every even integer $2n\geq4$ is the sum of two primes’ . This is what we know as Goldbach's Conjecture.
I am looking for a list/reference which explores the consequences of Goldbach's conjecture. Which problems will be proved?!
 A: Bratus and Pak have devised an algorithm which, given a black box (or "gray box") group $G$ which is known to be isomorphic to $S_n$ or $A_n$, actually explicitly finds such an isomorphism. The algorithm depends on certain quantitative versions of the Goldbach conjecture. They present two versions of the algorithms, one in which $n$ is given, and one in which merely an upper bound for $n$ is given.
A: Only a partial answer as it is too long for a mere comment. There are known connections between non vanishing of L-functions and some versions of the Goldbach conjecture. Gautami Bhowmik showed that some improvement on the error term for some summatory function of the number of Goldbach decompositions would lead to a tiny strip on the left of the vertical line $\Re(s)=1$ where the Riemann zeta function doesn't vanish.
Similarly, Goldston et al. proved that what they call weak Hardy-Littlewood-Goldbach conjecture would entail that no Landau-Siegel zero exists. There may also be some link between GRH and the upper bound for the quantity I denote by $\alpha_{n}$ in my question About Goldbach's conjecture, but it's hardly legible and I still lack a proper proof.
A: If the Goldbach's conjecture is true then for every even number $2n$ there exist numbers $n_1$ and $n_2$ such that
$\phi (n_1)+\phi (n_2)=2n$
Where $\phi$ is the $\phi$-Euler function.
