Maximum N that will hold this true Find the largest positive integer $N$ such that
$$\sqrt{64 + 32^{403} + 4^{N+3}}$$
is an integer
Is $N = 1003$?

First note that 
$$64 + 32^{403} + 4^{N+3} = 2^6 +2^{2015} + 2^{2N+6}.$$
It is easy to see that this cannot be a square of an integer if $2N+6 < 2015$, so now consider the way in which the expression can be a square:
$(2^3 + 2^{N+3})^2 = 2^6 + 2.2^3.2^{N+3} + 2^{2N+6} = 2^6 + 2^{N+7} + 2^{2N+6}$
Thus we need $N+7 = 2015$, so that $N=2008$.
