If n is a nonnegative integer, must 2^2n + 1 be prime? Prove or give a counterexample?

If that is 2^(2n) + 1, then all we need to do do disprove this is to find one example where it's false. So let's try a few numbers:

n | 2^(2n) + 1
-------------------
0 | 2^(2*0) + 1 = 2^0 + 1 = 1 + 1 = 2
1 | 2^(2*1) + 1 = 2^2 + 1 = 4 + 1 = 5
2 | 2^(2*2) + 1 = 2^4 + 1 = 16 + 1 = 17
3 | 2^(2*3) + 1 = 2^6 + 1 = 64 + 1 = 65 <-- not prime

So where n = 3, this is your counter-example showing that the statement is not true.

Assuming that means 2^(2n) + 1, let n=3. 2^6 = 64 + 1 = 65 is not prime.