exponential diophantine equation and its solutions

Question

The exponential Diophantine equation $x^2 + y^2 = 4x^n + 43$ has no integral solution $(x, y, z)$ for $n \geqslant 3.$ I have seen the problem in the lecture series in math conference. I do not know, how one can inspect the solutions of the cited above equation? We can check few solutions by trial and error. Here the condition is $n > 3$ or $n = 3$ case is failed to find solutions. If there is any mathematical proof to justify the statement? discuss.

Answer 1

The right hand side is $3\mod 4$, but this can never be the sum of two squares since any square is either $0^2\equiv 2^2 \equiv 0\mod 4$ or $1^2\equiv 3^2\equiv 1\mod 4$.