Although the analytical process used here includes some very long and involved algebraic derivations, the description of a "simple" proof is justified, because the technical level of the analysis is very elementary, being far below graduate level, and utilising only the very minimum amount of interpretative logic. It is easily of a level available to Fermat and his peers in their day.

In the same book margin that Fermat penned his famous conjecture, he also stated, [2],

I have discovered a remarkable proof of this theorem which the margins of this book are too small to contain."

It is now thought, [2], that Fermat's stated proof of the general case was flawed, and that he subsequently realised this, which may explain why he apparently did not again mention the conjecture, after challenging two of his peers to prove the cases n = 3 and n = 4, proofs for which he obviously must have known in order to issue the challenges.

Just before his death in 1665, Fermat penned a communication stating that all of his proofs used a technique that he himself devised known as "infinite descent". If this was the stated "remarkable proof" of his conjecture, then he would almost certainly have realised that it was flawed, because the method of infinite descent can only be successful when starting from a finite position. When starting from an infinite position, as proof of the Last Theorem would require, it must fail because even after an infinite number of descents, there would still be an infinite number to go, and it would therefore never reach a final result.

Also, in view of his communication above, it is also considered unlikely that Fermat's remarkable proof was similar to that presented here.

Finally, as is shown in the Appendix, Fermat's equation is but one version of a more general equation, some variations of which do exhibit integer solutions. It is not yet known whether there are others which, like Fermat's, only partially do so, i.e. as for n = 2.