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

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