There is FindInstance:
```
FindInstance[x^4 + y^4 + z^4 == t^4 , {x, y, z, t}, PositiveIntegers]
{{x -> 18796760, y -> 2682440, z -> 15365639, t -> 20615673}}
```
Minimize and Solveand a few other functions can also pop out AN answer.
If the problem is small enough, Solve with various constraints may do what you want.
Reduce can be useful for finding a possible algorithm to search for answers as well.
For deeper number theory functionality, it may be necessary to write your own algorithms.
