# Math Help - Diophantine Equation Proof

1. ## Diophantine Equation Proof

Show that the diophantine equation $x^4 - y^4 = z^2$ has no solutions in nonzero integers using the method of infinite descent.

2. Originally Posted by Aryth
Show that the diophantine equation $x^4 - y^4 = z^2$ has no solutions in nonzero integers using the method of infinite descent.
Assume that there is a positive integer solution, then pick one that has $z$ to be minimum. We have that $(x,z)=(y,z)=(y,z) = 1$ since otherwise we would be able to cancel by a common factor, contradicting that $z$ is mimimal. Now write $z^2 + (y^2)^2 = (x^2)^2$. This is a Pythagorean equation, there are two cases either $y$ is even or odd. If odd then $z=2ab,y^2=a^2-b^2,x^2=a^2+b^2$ for positive integers $a>b$ and $(a,b)=1$. But then $a^4 - b^4 = (a^2-b^2)(a^2+b^2) = (xy)^2$. Thus, we found another solution to this Diophantine equation which is even smaller then the original, thus we have a contradiction by infinite descent. Now you do the case when $y$ is even.