Does there exist a positive integer $k$ such that $A = 2 ^ k + 3 ^ k$ is a square number?

Question

Does there exist a positive integer $k$ such that $A = 2 ^ k + 3 ^ k$ is a square number? I have tried from $k = 1$ to $k = 10$, seeing that $A$ is not a square number. Could you please answer in the general case?

Answer 1

Note $A=2^k+3^k\equiv (-1)^k \pmod 4$, and a square number must be congruent to $0$ or $1$ modulo $4$, so $k$ must be even. Say $k=2m$.

Now $A=2^{2m}+3^{2m}=4^m+9^m\equiv(-1)^m+(-1)^m \pmod 5$, and a square number must be congruent to $0$ or $\pm1$ modulo $5$, so this is impossible.