@Zyxer I don't think it has. But you can try different trivial solutions by assuming x=y or x=w etc etc.

@Zyxer
Given \(x^2 + y^2 = 4z + 3\) with \((w, x, y, z)\in\Bbb{N}\), \(x\) is odd if and only if \(y\) is even. This results from the observation that \(4z + 3\) must be odd and the fact that, if \(k^n\) contains a factor of two for \((k, n)\in\Bbb{N}\), a factor of two must also necessarily be present in \(k\). It is not possible, then, for \(y^x = x^w\) to be true, because one side of the equation is an odd integer while the other is an even integer.