First, note that (a,b,n) = (2,1,1) is a solution since 2^2 + 1^4 = 5^1

now suppose (a,b,n) is a solution. compute (25a)^2 + (5b)^4:

= 5^4a^2 + 5^4b^4

= 5^4 * (a^2 + b^4)

= 5^4 * 5^n

= 5^(n+4)

Therefore, (25a, 5b, n+4) is also a solution. Since any solution can be used to generate a new solution with a larger n, the number of solutions is unbounded.

The solution generated from (2,1,1) is (50,5,5), the one after that is (1250, 25, 9), and the one after *that* is (31250, 125, 13). These first three are included in Larry's computer exploration. 

There's a similar infinite chain that starts from (3,2,2).

What I can't do is say whether or not these two chains are the complete set of solutions, only that they're both infinite.