It is shown that if a triple of distinct positive integers $(a,b,c)$ were to exist such that $a^n+b^n=c^n$ for some odd integer $ngeq 3$, then it must be Pythagorean, i.e. $a^2+b^2=c^2$ must hold too, from which a contradiction arises since this is possible only if either $a$ or $b$ are zero. We arrive at this conclusion by investigating the (partial) trace of a model hamiltonian operator whose energy levels correspond to those of the so-called H"uckel hamiltonian applied to rings containing an odd number of atomslying on a M"obius strip rather than a planar topology. Furthermore, the contradictory nature of our result implies the correctness of the associated statement contained in the famous Fermat's Last Theorem. Given the use of concepts from quantum mechanics and matrix algebra unknown at his time, and the fact that the essence of the present proof may not fit within a margin of a typical book, mystery still remains over Pierre de Fermat's {em demonstrationem mirabilem}.