example of Fermat’s last theorem

Fermat stated that for any $n>2$ the Diophantine equation $x^{n}+y^{n}=z^{n}$ has no solution in positive integers. For $n=4$ this follows from the following

Theorem 1.

$x^{4}+y^{4}=z^{2}$ has no solution in positive integers.

Proof.

Suppose we had a positive $z$ such that $x^{4}+y^{4}=z^{2}$ holds. We may assume $\gcd(x,y,z)=1$ . Then $z$ must be odd, and $x, y$ have opposite parity. Since $(x^{2})^{2}+(y^{2})^{2}=z^{2}$ is a primitive Pythagorean triple, we have

$x^{2}=2pq,y^{2}=q^{2}-p^{2},z=p^{2}+q^{2}$

(1)

where $p,q\in{\mathbb{N}}$ , $p<q$ are coprime and have opposite parity. Since $y^{2}+p^{2}=q^{2}$ is a primitive Pythagorean triple, we have coprime $s,r\in{\mathbb{N}}$ , $s<r$ of opposite parity satisfying

$q=r^{2}+s^{2},y=r^{2}-s^{2},p=2rs.$

(2)

From $\gcd(r^{2},s^{2})=1$ it follows that $\gcd(r^{2},r^{2}+s^{2})=1=\gcd(s^{2},r^{2}+s^{2})$ , which implies $\gcd(rs,r^{2}+s^{2})=1$ . Since $\left(\frac{x}{2}\right)^{2}=\frac{pq}{2}=rs(r^{2}+s^{2})$ is a square, each of $r,s,r^{2}+s^{2}$ is a square.

Setting $Z^{2}=q$ , $X^{2}=r$ , $Y^{2}=s$ $q=r^{2}+s^{2}$ leads to

$Z^{2}=X^{4}+Y^{4}$

(3)

where $Z^{2}=q<q^{2}+p^{2}=z<z^{2}$ . Thus, equation 3 gives a solution where $Z<z$ . Applying the above steps repeatedly would produce an infinite sequence $z>Z>z_{2}>\ldots$ of positive integers, each of which was the sum of two fourth powers. But there cannot be infinitely many positive integers smaller than a given one; in particular this contradicts to the fact that there must exist a smallest $z$ for which (1) is solvable. So there are no solutions in positive integers for this equation. ∎

A consequence of the above theorem is that the area of a right triangle with integer sides is not a square; equivalently, a right triangle with rational sides has an area which is not the square of a rational.

Title	example of Fermat’s last theorem
Canonical name	ExampleOfFermatsLastTheorem
Date of creation	2013-03-22 14:09:51
Last modified on	2013-03-22 14:09:51
Owner	Thomas Heye (1234)
Last modified by	Thomas Heye (1234)
Numerical id	9
Author	Thomas Heye (1234)
Entry type	Example
Classification	msc 11F80
Classification	msc 14H52
Classification	msc 11D41
Related topic	InfiniteDescent
Related topic	X4Y4z2HasNoSolutionsInPositiveIntegers