On equations $m^2+1=5^n$ I am looking for integer solutions of Diophantine equation $m^2+1=5^n$. I found that $m=0,n=0$ and $m=2,n=1$. I could not find any other solutions. I try to prove this but I could not. Could anyone help me to solve this equation?

Hint: By Mihăilescu's theorem the only solution to 
$$x^a - y^b = 1$$
where $a, b > 1$ and $x, y > 0$ is 
$$3^2 - 2^3 = 1$$ 
Now  you only have to consider the remaining cases where the exponents are not both more than $1$, and where the bases are not both more than $0$.
