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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 $4$ – Thomas Andrews Nov 9 '12 at 5:53
@ThomasAndrews!I got it. discuss the above post mathematically by taking the cases. – vmrfdu123456 Nov 9 '12 at 6:42

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 $1^2\equiv 3^2\equiv 1\mod 4$.

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@pi367!fine. Thank U – vmrfdu123456 Nov 10 '12 at 6:03

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