Regarding to our hypothesis in <https://math.stackexchange.com/questions/1918406/a-hypothesis-about-the-conjecture-every-even-number-is-the-difference-of-two-p> , we guess that the following statements are equivalent (is it true?):


**(a)** For every integer number $N$ there exist prime numbers
$p\_1,p\_2,p\_3,p\_4$ such that $N=p\_1+p\_2-p\_3-p\_4$;


**(b)** For every distinct non-zero integers $a, b$, at least one of the numbers $a, b$ and $a-b$ can be expressed as the difference of two primes.



Two equivalent statements about primes
Statement a) is true for all $N$. This follows from the fact that the number of even integers which cannot be expressed as the sum of 2 primes is small. The best result in this direction is due to Pintz, who showed that there are $\mathcal{O}(x^{2/3})$ even integers below $x$, which cannot be written as a sum of two primes. In particular there is some $x\_0$, such that for $x>x\_0$ the number of exceptions is $<x^{3/4}$. Note however, that for the following argument much weaker estimates suffice.


Assume first that $N$ is even. If $N$ cannot be written as the difference of two Goldbach numbers $<x$, then for all $n<x-N$ we have that at most one of $n, n+N$ is Goldbach. Hence the number of Goldbach numbers below $x$ is at most $\frac{x-N}{4}+N$, which for $x>\max(x\_0, 5N)$ gives a contradiction. 


Next suppose that $N$ is odd. Then take $p\_4=2$. If $N+2$ cannot be written in the form Goldbach minus prime, where the Goldbach number is bounded by $x$, then for each prime $p\leq x-N-2$ we have that $N+2+p$ is not Goldbach. Hence the number of even integers below $x$, which are not Goldbach, is at least $\pi(x-N-2)$. Again taking $x=\max(x\_0, 4N)$ gives a contradiction. 


