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First steps in general proofs

In contrast with the rapid development of number theory, the proof of FLT advanced very slowly, as the proof of each special case was more difficult than its previous. Evidently there was the need to find a new way to face the problem. So did Lamé in the beginning of 1847. By transforming the expression xn+yn into a multiplication of n factors, using algebraic numbers different from those used by his predecessors and a tactic making the use of these numbers more essential, Lamé presented to the members of the French Academy of Sciences a general proof of FLT. The excited Lamé did not take the whole glory to himself and mentioned that he came to this result after discussing it with his colleague Joseph Liouville. Immediately after the presentation, Liouville pointed out an important element that was missing in the proof: there is a great resemblance between the integers and the algebraic numbers used by Lamé, a resemblance that enables to learn about the properties of algebraic numbers from those of the integers, but to complete the proof one must show that a theorem similar to the fundamental theorem of arithmetic is valid also to the algebraic numbers.


In the weeks to follow, Lamé, and concurrently to him Cauchy, who has a central role in the life of first-year mathematics students, tried to find the missing proof, but in vain. "If only you were in Paris or I was in Berlin, all of this would be avoided", wrote miserable Lamé to his friend Dirichlet. A proof that Lamé's mistake is not reparable was found three years earlier by Ernest Edward Kummer, who later replaced Dirichlet in the University of Berlin, but he published it in an unknown journal, so Lamé did not get it on time. After noticing the story about Lamé and Liouville, Kummer sent a copy of his proof to Liouville, who brought it to public knowledge by publishing it again in a journal he edited (in this journal appeared also the wrong proof of Lamé). Euler had made a mistake similar to that of Lamé, so we must ask ourselves whether Fermat had made a similar mistake too when he wrote in the margin of the page about his marvelous proof.


Kummer tackled the problem that caused Lamé's failure while working on Reciprocity Law, a subject in number theory that was created by Gauss, who expressed the hope that the advancement of the subject will enable a simple proof of FLT. To overcome the problem Kummer developed the ideal numbers, where the fundamental theorem of arithmetic is valid, making them useful in solving many problems in number theory. The importance of this idea steps out of the scope of number theory, and it was broadened by Richard Dedekind to a separate mathematical branch - Ideal Theory.

Using ideal numbers for handling FLT, in a strategy not different from that of Lamé, enabled Kummer to make the main breakthrough, until recent years, in the proof of FLT. A few weeks after Lamé's failure, Kummer proved FLT not as to a specific power, but as to a wide group of powers, named by him as "regular primes". To show that FLT is valid for some n, it is enough to ensure, by calculation, that this n is a regular prime. In the following four years Kummer calculated that among all primes less than 100, only 37, 59 and 67 are not regular. In 1857, Kummer found more conditions to show that a power conforms FLT. These conditions completed the proof of FLT to all powers less than 100.

© David Shay, 2003



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