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.