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Is this the proof we were expecting?

FLT had gone a long way during the years between the day it was scribbled by Fermat to the day it was proved by Wiles. From a conjecture stated by an amateur in his spare time, to a proof that required total commitment for years. From a short and simple conjecture that everyone can understand, to a long and complex proof that only experts can understand. All mathematics has gone through the same change during that period, and one may doubt whether it has not exceeded reasonable complexity. Not only when a proof is based on hundreds of computer hours, but also when the proof is produced by using the classical tools of pencil and paper - who can check a proof hundreds of pages long?

In an interview made a short time after he presented his proof, Wiles expressed his hope that a simpler proof would be found. Wiles himself realized this hope while he was amending his proof and shortening it by a quarter of its original size. This process may be encouraged by Mordell, which compared mathematical research to mountain climbing - the first conquest of the mountain's peak is a hard and demanding task, but the next climbing to the same peak is easier. Another peak was conquered in 1999, when four mathematicians, including Richard Taylor, published a proof for the full Taniyama-Shimura conjecture. This proof, which is considered one of the major achievements of twentieth century mathematics, is based on Wiles' proof.

Proving FLT does not solve Fermat's riddle: did Fermat have a marvelous proof to his theorem, or was he wrong when he thought he had such a proof? Here is the riddle as Mordell posed it in 1921:

"Is it probable, that nearly three centuries ago, Fermat really proved this theorem, which still baffles mathematicians who have at their disposal the wonderful and far reaching developments in mathematics since Fermat's time - especially as it seems likely that Fermat's methods could only be elementary considered from a modern standpoint? From what is known of Fermat's character, it is fairly certain that at any rate he was under the impression that he had a proof meriting his description of it".

Wiles' proof is based on modern mathematical tools, which surely were not in the hands of Fermat. Various signs hint that Fermat himself noticed that his marginal note was wrong, but searchers of glory are still trying to find - now with more confidence, since the doubt that maybe there is no proof at all has been removed - a proof that is not beyond seventeenth century mathematics (Wiles tried to do that in his boyhood). Roger Apéry of Caen University in France came to such an achievement in 1978, when he succeeded to solve an unsolved problem stated in the eighteenth century by Euler, using only mathematical knowledge known in Euler's period. Weil's opinion, as stated in his book Number Theory - An approach through history may cool the enthusiasm to deal at this way with FLT: "There can hardly remain any doubt that this was due to some misapprehension on his part, even though, by a curious twist of fate, his reputation in the eyes of the ignorant came to rest chiefly upon it".

Is it possible that FLT has a simple solution that till now slipped away from everybody? The following curiosity, in which many of the heroes of our story are involved, shows that such case is not impossible. Amicable numbers are pair of integers that each of them is equal to the sum of the divisors of the other. Pythagoras knew the smallest amicable numbers, 220 and 284, and saw them as a symbol of friendship among people (the divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, and their sum is 284; the divisors of 284 are 1, 2, 4, 71, 142, and their sum is 220). Though Pythagoras and his students, and many generations after, gave symbolic importance to amicable numbers, only in 1636 Fermat found another pair of such numbers, 17,296 and 18,416, and Descartes found some time later a third pair, 9,363,584 and 9,437,056. In the eighteenth century, Euler published a list of 64 pairs of amicable numbers (two of these pairs were found to be wrong), and Legendre succeeded to find one more pair in 1830. Surprisingly, in 1867 a 16-year-old Italian, Nicolò Paganini, found a pair that evaded everybody, 1,184 and 1,210, which is the second by size, on the list of amicable numbers.

Number theory is one of the "purest" branches of mathematics, namely it is not rich in useful results for daily life. What is the reason, under these circumstances, for the effort to prove Fermat's Last Theorem? An answer to this question may be found in Siméon-Denis Poisson's words in the funeral of Legendre: "Questions concerning the properties of the numbers, far away from any practical use, have a single attraction power, but a very powerful one: they are extraordinarily hard".


© David Shay, 2003



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