Fermat's last theorem
A challenge for many long ages
Fermat's last theorem is a conjecture put forward by Pierre de Fermat in 1637 in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The note was found after his death, and the original is now lost. However, a copy was included in the appendix to a book published by Fermat's son. Fermat's note read:
It is impossible to write a cube as a sum of two cubes, a fourth power as a sum of fourth powers, and, in general, any power beyond the second as a sum of two similar powers. For this, I have found a truly wonderful proof, but the margin is too small to contain it.Fermat claimed that the Diophantine equation xn + yn = zn has no integer solutions for n > 2. It turns out he was right. But the proof had to wait 350 years and involved such advanced techniques, virtually none of which existed in the 17th century, that it seems very unlikely that Fermat really had found an elementary proof.
Fermat's last theorem – now truly a theorem – was finally proved correct by Andrew Wiles in 1994.1 In order to reach that dizzy height, however, Wiles had to draw on and extend several ideas at the core of modern mathematics. In particular, he tackled the Shimura-Taniyama-Weil conjecture, which provides links between the branches of mathematics known as algebraic geometry and complex analysis. This conjecture dates back to 1955, when it was published in Japanese as a research problem by the late Yutaka Taniyama. Goro Shimura of Princeton and Andre Weil of the Institute for Advanced Study provided key insights in formulating the conjecture, which proposes a special kind of equivalence between the mathematics of objects called elliptic curves and the mathematics of certain motions in space. Interestingly, the Wiles proof of Fermat's last theorem was a by-product of his deep inroads into proving the Shimura-Taniyama-Weil conjecture. Now, the Wiles effort could help point the way to a general theory of three-variable Diophantine equations. Historically, mathematicians have always had to state and solve such problems on a case-by-case basis. An overarching theory would represent a tremendous advance.
Related entry ABC conjecture
Related category NUMBER THEORY
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