Reply to thread

Message: <blockquote data-quote="mag123" data-source="post: 19523727" data-attributes="member: 73975">Fermat wrote this marginal note around 1630I have discovered a truly remarkable proof which this margin is too small to contain. Fermat almost certainly wrote the marginal note around 1630, when he first studied Diophantus's Arithmetica. It may well be that Fermat realised that his remarkable proof was wrong, however, since all his other theorems were stated and restated in challenge problems that Fermat sent to other mathematicians. Although the special cases of n = 3 and n = 4 were issued as challenges (and Fermat did know how to prove these) the general theorem was never mentioned again by Fermat. (xq)p + (yq)p = (zq)p. <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Euler.html" target="_blank">Euler</a> wrote to <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Goldbach.html" target="_blank">Goldbach</a> on 4 August 1753 claiming he had a proof of Fermat's Theorem when n = 3. However his proof in Algebra (1770) contains a fallacy and it is far from easy to give an alternative proof of the statement which has the fallacious proof. There is an indirect way of mending the whole proof using arguments which appear in other proofs of <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Euler.html" target="_blank">Euler</a> so perhaps it is not too unreasonable to attribute the n = 3 case to <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Euler.html" target="_blank">Euler</a>.Despite large prizes being offered for a solution, Fermat's Last Theorem remained unsolved. It has the dubious distinction of being the theorem with the largest number of published false proofs. For example over 1000 false proofs were published between 1908 and 1912. The only positive progress seemed to be computing results which merely showed that any counter-example would be very large. Using techniques based on <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Kummer.html" target="_blank">Kummer</a>'s work, Fermat's Last Theorem was proved true, with the help of computers, for n up to 4,000,000 by 1993. This, however, is not the end of the story. On 4 December 1993 Andrew Wiles made a statement in view of the speculation. He said that during the reviewing process a number of problems had emerged, most of which had been resolved. However one problem remains and Wiles essentially withdrew his claim to have a proof. The key reduction of (most cases of) the Taniyama-Shimura conjecture to the calculation of the Selmer group is correct. However the final calculation of a precise upper bound for the Selmer group in the semisquare case (of the symmetric square representation associated to a modular form) is not yet complete as it stands. I believe that I will be able to finish this in the near future using the ideas explained in my Cambridge lectures.  <a href="http://www-history.mcs.st-and.ac.uk/HistTopics/Fermat's_last_theorem.html" target="_blank">http://www-history.mcs.st-and.ac.uk/HistTopics/Fermat's_last_theorem.html</a>   On 6 October <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Wiles.html" target="_blank">Wiles</a> sent the new proof to three colleagues including <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Faltings.html" target="_blank">Faltings</a>. All liked the new proof which was essentially simpler than the earlier one. <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Faltings.html" target="_blank">Faltings</a> sent a simplification of part of the proof. No proof of the complexity of this can easily be guaranteed to be correct, so a very small doubt will remain for some time. However when Taylor lectured at the British Mathematical Colloquium in Edinburgh in April 1995 he gave the impression that no real doubts remained over Fermat's Last Theorem. </blockquote>

Verification: Winadiyakata thappara keeyak tibeda?

Top Bottom