It is a shame that Andrew Wiles spent so many of the prime years of his life following such a difficult path to proving Fermat's Last Theorem, when there exists a much shorter and easier proof. Indeed, this concise, elegant alternative, reproduced below, is almost certainly the one that Fermat himself referred to in the margin of his copy* of Bachet's translation of Diophantus's Arithmetica--how arrogant of future chroniclers of mathematical history to insinuate that the garlicky Gallic genius was incapable of formulating the crucial insight! Why, a child can follow the logic below...however, for the benefit of those readers who are no longer children, a short set of notes fleshing out the argument is provided after the proof.
The Theorem: xª + yª = zª has no positive integer solutions (x, y, z, a) for a > 2. (Pierre De Fermat, 1601-1665)
I) At least one of the following two sentences is true.
II) The preceding sentence is false.
III) xª + yª = zª has no positive integer solutions (x, y, z, a) for a > 2.
A. Statement I is either
true or false.
B. Assume I is true. Then so is either II or III. But II is false, as it denies the truth of I. Hence III must be the true statement of the two.
C. Assume I is false. Then both II and III must not be true. But II agrees that I is false, so II is true. This is a contradiction.
D. Since assuming I is false leads to a contradiction, I is true.
E. Since I is true, so is III (see note B.) Thus III, Fermat's Last Theorem, must hold.
without saying that the system employed above is capable of great generalization.
But mathematicians are a stubborn lot, and, despite the efficiency and
aesthetic appeal of the approach, legions of haunted, driven men and women
will continue to pursue arcane mathematical truths by means of tortuous,
convoluted, labyrinthine arguments--and that's OK, because it keeps them
busy and off the streets, where their generally preoccupied state dramatically
increases their probability of being killed by drivers using cell-phones.
But even I have to admit that an over-used method is a life-essence-draining method, and have proceeded from proofs utilizing the algorithm above to even more startling applications of easily-overlooked syllogistic constructions. In fact, I have discovered a truly remarkable proof of Goldbach's conjecture which this web page is too small to contain...
Some historians have claimed that the copy in question originally belonged to Pierre's twin sister, Polly, and have advanced the dubious theory that she scrawled the marginalia as a joke on her admittedly somewhat pompous brother, whom she outperformed in math up until 7th grade, when she discovered garcons. Although one chemical analysis does support the possibility that the message was written with an eyebrow pencil, skepticism is advised. Polly Fermat did, however, strike an early blow for feminism when she refused to change her last name upon marrying her childhood sweetheart, Jean-Jacques Nomial.
to Home Page