example of Fermat’s last theorem

Fermat stated that for any n>2 the Diophantine equationMathworldPlanetmath xn+yn=zn has no solution in positive integers. For n=4 this follows from the following

Theorem 1.

x4+y4=z2 has no solution in positive integers.


Suppose we had a positive z such that x4+y4=z2 holds. We may assume gcd(x,y,z)=1. Then z must be odd, and x,y have opposite parity. Since (x2)2+(y2)2=z2 is a primitive Pythagorean tripleMathworldPlanetmath, we have

x2=2pq,y2=q2-p2,z=p2+q2 (1)

where p,q, p<q are coprimeMathworldPlanetmath and have opposite parity. Since y2+p2=q2 is a primitive Pythagorean triple, we have coprime s,r, s<r of opposite parity satisfying

q=r2+s2,y=r2-s2,p=2rs. (2)

From gcd(r2,s2)=1 it follows that gcd(r2,r2+s2)=1=gcd(s2,r2+s2), which implies gcd(rs,r2+s2)=1. Since (x2)2=pq2=rs(r2+s2) is a square, each of r,s,r2+s2 is a square.

Setting Z2=q, X2=r, Y2=s q=r2+s2 leads to

Z2=X4+Y4 (3)

where Z2=q<q2+p2=z<z2. Thus, equation 3 gives a solution where Z<z. Applying the above steps repeatedly would produce an infiniteMathworldPlanetmath sequenceMathworldPlanetmath z>Z>z2> 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 triangleMathworldPlanetmath 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