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

: X n + Y n = Z n .

Therefore in terms of the affine plane its equation is

: x n + y n = 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 (mathematics)

:( 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.