TITLE: Does $4x^2+1=5^y$ have a solution in integers with $y>1$?
QUESTION [5 upvotes]: Consider the following equation :

$4x^2+1=5^y~$ with $y>1$

Has this equation solutions in integers ?
I wrote small Maple program in order to find solutions but couldn't find anyone .
for x from 1 to 6000 do
for y from 2 to 2000 do
if 4*x^2+1 = 5^y then
print(x,y);
end if;
end do;
end do;

REPLY [5 votes]: Not according to Catalan's conjecture (actually a theorem), since you have $5^y-(2x)^2=1$.