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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; 

1 Answers 1

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