Limit to solution to Integers

Prime is not suitable for solving Diophantine equations. The algorithm used in the solve block needs to be able to change the variables in small steps and see if the overall error gets smaller or not. If you add constraints like the ones shown in the picture below, small changes will not alter the error and so Prime gives up.

You may use the symbolic solver, but it only finds the trivial solution 0/0/0. And demanding it to give all solutions (modifier "fully") gives a rather useless result.

So in case of your task, a brute force attack seems to be the best approach.

In case you also want solutions with negative integers, you should filter the solutions containing a zero as otherwise you would see too many solutions of the form -5/0/5.