I believe this is Fermat's Last Theorem, of which there are several solutions (e.g., Andrew Wiles' solution, 1993). The maximum number of characters allowed for answers on this web site is 8000 (due to restrictions on the database server) and even if I knew the shortest solution, I doubt I could express it in such a small number of characters and without diagrams! So please look up references to this theorem on the Internet (e.g., www.fermatproof.com). That's what I eventually did!