de Polignac conjecture: For any positive even number $d≥2$, there are infinitely many prime gaps of size $d$.
We know (https://mathoverflow.net/questions/339781/generalization-of-wilsons-theorem-for-prime-tuples) that $(k,k+2)$ forms a twin prime pair if and only if $((4(k-1)!+k+4)/(k(k+2)))$ is a positive integer.
My question is about finding a similar criterion for the de Polignac conjecture by using the Wilson theorem (https://en.wikipedia.org/wiki/Wilson%27s_theorem).