I assume given x and y,you need to find t.

First observe following trivial solutions:

1.t=1 for any x and y is a solution.

2.If x and y both are divisible by 13,then any t will satisfy given conditions.

Consider the case when both x and y leave same remainder when divided by 13.In this case any t=12k+1 where k is non-negative integer,satisfies given condition.(Using Fermat's Little Theorem).

For the case x and y leave different remainders,I could conclude that you should only look for t in the set {5,7,11,..}(Set of all numbers whose square leave remainder 1 when divided by 12).I think(?) there does not exist a solution in this case.