Originally Posted by

**alexmahone** Let $\displaystyle a = 2m$.

$\displaystyle 5^b=t^2-2^{2m}=(t+2^m)(t-2^m)$

Then,

$\displaystyle t+2^m=5^p$

$\displaystyle t-2^m=5^q$

where p and q are non-negative integers.

Assume that p and q are both positive integers.

Then, $\displaystyle 2t=5^p+5^q\equiv 0\ (mod\ 10)$

$\displaystyle t\equiv 0\ (mod\ 5)$

Consider the equation $\displaystyle t+2^m=5^p$.

$\displaystyle 5 | t$ and $\displaystyle 5 | 5^p$

So, $\displaystyle 5 | 2^m$ (Contradiction)

So, either p or q is zero. (Both p and q cannot be zero, because then b would be zero, which is not allowed by the question.)