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Thread: Diophantine Equation Proof

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    Super Member Aryth's Avatar
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    Diophantine Equation Proof

    Show that the diophantine equation $\displaystyle x^4 - y^4 = z^2$ has no solutions in nonzero integers using the method of infinite descent.
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    Quote Originally Posted by Aryth View Post
    Show that the diophantine equation $\displaystyle x^4 - y^4 = z^2$ 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 $\displaystyle z$ to be minimum. We have that $\displaystyle (x,z)=(y,z)=(y,z) = 1$ since otherwise we would be able to cancel by a common factor, contradicting that $\displaystyle z$ is mimimal. Now write $\displaystyle z^2 + (y^2)^2 = (x^2)^2$. This is a Pythagorean equation, there are two cases either $\displaystyle y$ is even or odd. If odd then $\displaystyle z=2ab,y^2=a^2-b^2,x^2=a^2+b^2$ for positive integers $\displaystyle a>b$ and $\displaystyle (a,b)=1$. But then $\displaystyle a^4 - b^4 = (a^2-b^2)(a^2+b^2) = (xy)^2$. 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 $\displaystyle y$ is even.
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