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Encyclopedia > Fermat curve

In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation

Xn + Yn = Zn.

Therefore in terms of the affine plane its equation is

xn + yn = 1,

with integer solutions (not all zero) of the projective equation corresponding to rational number solutions of the affine equation. Proportional integer solutions correspond to the same rational solution. This curve therefore can be used to formulate Fermat's Last Theorem in a geometric way (hence its name).


The Fermat curve is non-singular and has genus

(n − 1)(n − 2)/2.

This means genus 0 for the case n = 2 (a conic) and genus 1 only for n = 3 (an elliptic curve). The Jacobian variety of the Fermat curves has been studied in depth.


  Results from FactBites:
 
PlanetMath: Fermat's last theorem (582 words)
Fermat's last theorem was actually a conjecture and remained unproved for over 300 years.
We cannot imagine how Fermat's last theorem could be proved without these advanced mathematical tools, which include group theory and Galois theory, the theory of modular forms, Riemannian topology, and the theory of elliptic equations.
Assuming Fermat's teaser was truthful, and Fermat was not in error, this apparent paradox has led some to jokingly attribute supernatural abilities to Fermat.
  More results at FactBites »


 

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