I need to find pairs of primes that fit the equation that $x=(y \times c)+1$, where $c$ is a positif integer. or you can say that $x\equiv 1\pmod y$
is there a criterion in finding these prime pairs, or is it just coincidences. if so then what are the examples.
As pointed out in a comment above, if you first fix the larger prime $x$, it is easy to compute the possible values for $y$, so long as you can compute a prime factorization.
On the other hand, if you start with the smaller prime $y$ and try to search for $x$, it is a theorem that there are infinitely many primes in the list $1, 1+y, 1+2y, 1+3y,\ldots$ which are possible choices of $x$. (Namely, this is a special case of Dirchlet's Theorem on arithmetic progressions.) So in that sense the criterion is that it is always possible to find such a pair.
I'm not sure what else you would want for a criterion. Maybe you want to know how long it takes to find the first prime - this is probably a bit up to "coincidences" as you say, but there are approximations available, generalizing the prime number theorem. You can read about the prime number theorem for arithmetic progressions (Wikipedia) or the Siegel-Walfisz theorem (Wikipedia).