We know that Wilson's theorem states the following :

$x$ is a prime if $(\frac {\Gamma(x)+1}{x})$ is an integer .

We can extend this to Twin primes as :

$x$ and $x+2$ is prime if $(\frac {4(\Gamma(x)+1)+x}{x(x+2)})$ is an integer.

Now my question :

what is an equivalent result to the tuples of primes ?

i.e.

for given $k_0$

${\mathcal H} = (h_1, \ldots ,h_{k_0})$

and $n+ {\mathcal H } $ consists entirely of primes.

What is equivalence in terms of analog of Wilson's theorem to State the above condition ( I know it's relatively easy ) ?

(For twin primes ${\mathcal H} = (0,2)$)

Can we first prove the infinity of primes by proving the statement there are infinitely many integers of the form $(\frac {\Gamma(x)+1}{x})$ if x is an integer ; and then proceed in similar direction for tuples ? -- actually this is my main question .

( Forgive my English and type of typing I'm new here )