05B Fermat's last theorem

# 05B Fermat's last theorem - The URL of this page is...

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The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Fermat's_last_theorem.html Fermat's last theorem Pierre de Fermat died in 1665. Today we think of Fermat as a number theorist, in fact as perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was in fact a lawyer and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous article written as an appendix to a colleague's book. There is a statue of Fermat and his muse in his home town of Toulouse. Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son, Samuel undertook the task of collecting Fermat 's letters and other mathematical papers, comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way the famous 'Last theorem' came to be published. It was found by Samuel written as a marginal notein his father's copy of Diophantus 's Arithmetica . Fermat's Last Theorem states that x n + y n = z n has no non-zero integer solutions for x , y and z when n > 2. Fermat wrote I 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 . In fact in all the mathematical work left by Fermat there is only one proof. Fermat proves that the area of a right triangle cannot be a square. Clearly this means that a rational triangle cannot be a rational square. In symbols, there do not exist integers x , y , z with x 2 + y 2 = z 2 such that xy /2 is a square. From this it is easy to deduce the n = 4 case of Fermat 's theorem. It is worth noting that at this stage it remained to prove Fermat 's Last Theorem for odd primes n only. For if there were integers x , y , z with x n + y n = z n then if n = pq , ( x q ) p + ( y q ) p = ( z q ) p . Euler wrote to Goldbach 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 Euler so perhaps it is not too unreasonable to attribute the n = 3 case to Euler .

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The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Fermat's_last_theorem.html Euler 's mistake is an interesting one, one which was to have a bearing on later developments. He needed
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