Find out the pairwise prime integer solutions?

Question

Number of pairwise prime positive integer solutions of $(x,y,z)$ in the equation $$(1/x^2) + (1/y^2) = (1/z^2)$$

I had started the problem with taking numbers but unsuccessful to reach towards solution. Is there any general way to reach solution with ease!

user10354138 · Answer 1 · 2020-07-21T19:23:34.607

3

Note that we have $$ \frac1{x^2}+\frac1{y^2}=\frac1{z^2}\iff (xz)^2+(yz)^2=(xy)^2 $$ so $z\mid xy$. But we want $x,y,z$ pairwise coprime, so $z=1$. Thus we have $x^2y^2=x^2+y^2$, or equivalently, $(x^2-1)(y^2-1)=1$ which has no solutions in $\mathbb{Z}^+$.

edited Jul 21 '20 at 19:23

answered Jul 21 '20 at 19:10

user10354138

33,239

Sorry, what is meant by z| xy ? – Jul 21 '20 at 19:14
$z$ divides $xy$. – user10354138 Jul 21 '20 at 19:15
@Sil that is not pairwise coprime --- $\gcd(x,y)=5\neq 1$ for example. – user10354138 Jul 21 '20 at 19:34
Oh right, obviously, i have looked at something else (corresponding pythagorean triple is pairwise coprime) – Sil Jul 21 '20 at 19:36

Find out the pairwise prime integer solutions?

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