# A counterexample to $x^n + y^n = h^2 + nf^2$ implies $x + y = h'^2 + nf'^2$ in the integers

In the same 1807 letter, Sophie claimed that if $x^n + y^n$ is of the form $h^2 + nf^2$, then $x + y$ is also of that form. Gauss replied with a counterexample: $15^{11} + 8^{11}$ can be written as $h^2 + 11f^2$, but $15 + 8$ cannot.
WolframAlpha confirms that the equation $15^{11} + 8^{11} = h^2 + 11f^2$ has two solutions in the positive integers, $(h, f) = (935166, 841201)$ and $(h, f) = (1595826, 745391)$, and it is obvious that $15 + 8 = h^2 + 11f^2$ has no solutions in the integers.
However, how would one be able to confirm that $15^{11} + 8^{11} = h^2 + 11f^2$ has solutions in the positive integers without such a computational aid? In a related question, how might Gauss have come up with this counterexample in the first place?