Find all $(x,y) \in \mathbb{N} \times \mathbb{N}$ such that $5^{x}+3^{y}$ is a perfect square

Question

• $\textbf{Question.}$ Find all $(x,y) \in \mathbb{N} \times \mathbb{N}$ such that $5^{x}+3^{y}$ is a perfect square

• One thing which I observed is the following. Since $5 \equiv 1 \pmod{4}$, this says $5^{x}\equiv 1 \pmod{4}$. Now if $y$ is even, then $3^{y}\equiv 1\pmod{4}$, then $5^{x}+3^{y}\equiv 2 \pmod{4}$. But there are no squares which are $2\pmod{4}$. This says $y$ can't be even. Hence $y$ is odd.

• Not sure how to proceed further. Any help would be deeply appreciated.

Answer 1

As the OP noted, $y$ has to be odd.

Every perfect square is congruent to 0, 1, or 4 modulo 5, hence if $x>0$, then $3^y \equiv 1,4 \pmod 5$, which implies that $y$ has to be even. Therefore there are no solutions with $x>0$.

For $x=0$, the equation becomes $1+3^y = m^2$ for some positive integer $m$. Then $(m-1)(m+1) = 3^y$. Since 3 can't divide both factors, the only possibility is $m-1=1$ and $m+1=3^y$. The only solution is $m=2$, when $y=1$ as well.

Hence $(x,y) = (0,1)$ is the unique solution.