Math Through the Ages by Berlinghoff William P.; Gouvea Fernando Q.; & Fernando Q. Gouvêa

Author:Berlinghoff, William P.; Gouvea, Fernando Q.; & Fernando Q. Gouvêa
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2019-02-26T16:00:00+00:00

Lamé presented his proof to the Paris Academy. Joseph Liouville was suspicious. Why should the new numbers behave like the old ones? Knowing that German mathematician Ernst Kummer had worked with similar ideas, he wrote Kummer askingabout this. Kummer replied that he had known for a while that the new numbers did not have the required unique factorization property, so Lamé's proof did not work. But Kummer got interested in the problem, and he ended up proving a far-reaching result. He identified a nice property that many prime numbers have, and called such primes "regular.'' Then he gave a proof of Fermat's Last Theorem for any exponent n which is a regular prime. This doesn't cover all the primes, but it does cover a great many. And Kummer's ideas led to further advances that allowed mathematicians to extend his method to other primes, as well.

Nevertheless, for a long time after Kummer there was little progress towards a general proof. In 1909 Paul Wolfskehl, a wealthy German mathematician, established a prize fund of 100,000 marks to reward whoever could find a proof. The prize generated many more failed proofs before it disappeared with the collapse of the German economy after World War I.2 For decades after that, the prospect of fame alone was enough to generate many more erroneous attempts, but a real, correct proof still seemed very distant.

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