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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.

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    Shouldn't have any solutions for any $n$ since $x^2+y^2-3$ is never divisible by $4$2012-11-09
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    @ThomasAndrews!I got it. discuss the above post mathematically by taking the cases.2012-11-09

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