## Fermat's last theoremA challenge for many long ages 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 x + ^{n}y = ^{n}z
has no integer solutions for ^{n}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 byproduct 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. ## Reference- Wiles, A. "Modular Elliptic-Curves and Fermat's Last Theorem. "
*Ann. Math.*, 141: 443-551, 1995.
## Related entry• ABC conjecture## Related category• NUMBER THEORY | |||||

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