The Fermat curve of order *n* is the set of points satisfying

*x*^{n} + *y*^{n} = 1

for a positive integer *n*. Fermat’s last theorem is equivalent to saying there are no non-trivial rational points on the Fermat curve of order *n* > 2. (The trivial points have *x* or *y* equal to 0.)

## Parameterization

The Fermat curve of order 2 is a circle, which can be parameterized by (cos *t*, sin *t*) for 0 ≤ *t* < 2π.

The Dixon elliptic functions cm and sm, mentioned in the previous post, satisfy

cm^{n} *x* + sm^{n} *y *= 1

and so the Fermat curve of order 3 can be parameterized by (cm *t*, sm *t*). What is the necessary range on *t*?

## Period

In the previous post we said that the function sm maps the unit circle to an equilateral triangle. One of the vertices of this triangle is

*v* = ⅓ *B*(⅓, ⅓)

in the complex plane, and the other two vertices are rotations of this vertex about the origin. We could have defined *v* by

*v* = π_{3} / 3

where

π_{3} = *B*(⅓, ⅓).

The Dixon functions are periodic with period π_{3 [1]} and we can take for our parameterization −⅓ π_{3} < *t* < ⅔ π_{3}.

## Plot

Here’s a plot of the Fermat curve of order 3.

In the first quadrant, the Fermat curve is a squircle.

The equation of a squircle is the same as that of a Fermat curve, except you take the absolute value of the arguments. The exponent is usually denoted *p* rather than *n* and can be any real number *p* > 2 rather than a positive integer.

## Related posts

[1] Because the Dixon functions are elliptic functions, they are periodic in *two* independent directions. The Dixon functions are also periodic in the direction exp(2π*i*/3), and the period is also of length π_{3} in that direction. Elliptic functions can have different period lengths in different directions, but for the Dixon functions the two period lengths are the same.