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From the Margins to Glory: the Story of Andrew Wiles
19.03.2021 Manuel de León
Mathematics
Andrew Wiles

Solving Fermat’s Last Theorem is one of the most exciting episodes in the history of mathematics. Now the Norwegian Academy of Science and Letters has awarded the Abel Prize to the mathematician who put an end to the puzzle: Andrew Wiles.

Once upon a time there was a French notary who was an amateur mathematician and loved to write in the margins of the pages of his books. In one of them, Diophantus’s Arithmetica, he left a note that intrigued successive generations of mathematicians for three and a half centuries. He scribbled a simple observation about integers: a power, other than a square, cannot be broken down as the sum of the powers of the same order; or, in other words, that the equation xn + yn = zn does not support integer solutions for n>2 (for n=2, it is possible to find solutions, these are the so-called Pythagorean triples). He added: “I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.” Unfortunately, the notary, Pierre de Fermat, did not leave any clues to this demonstration in any other document, and for three centuries, the most brilliant mathematicians tried to find it.

ALGEBRAIC GEOMETRY AND ELLIPTIC CURVES

It took a radically different approach that was provided by two Japanese mathematicians, Yutaka Taniyama and Goro Shimura. They formulated a conjecture on a seemingly unrelated branch, number theory where the problem is included: algebraic geometry, and more specifically the so-called elliptic curves (which, incidentally, are already beginning to be used in modern cryptography).

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