Extending the plane trigonometric proof of Fermat’s Last Theorem to the case n = 3

We extend the plane trigonometric approach that we used to prove the case $n = 4$ of Fermat's Last Theorem, to the case $n=3.$ We show that all real positive triplets satisfying $a^\phi + b^\phi = c^\phi$ for $\phi > 1$ are triangles. As in the case of $n=4,$ we equate the Pythagorean and Fermat descriptions of the triangles for a particular smaller side while fixing the other sides, with $\phi=n$ being any positive integer. We hence show the existence of Ferma<span class="fontstyle0">t–</span>Pythagoras polynomials for $n \ge 3.$ For the case $n=3,$ we explicitly derive an analytic expression for the roots of the polynomials. We prove from this expression that the real roots, which are equal to the length of the sides, are irrational.