Is it true that for every rational $q \geq 3$ , the following equation has a solution over $\mathbb N$ ?

$$q=\frac{x}{y} +\frac{y}{z} + \frac{z}{x}$$

  • $\begingroup$ I couldn't solve it even for particular case $q=4$ :( $\endgroup$ – Mahan Feb 26 '12 at 16:00
  • 1
    $\begingroup$ By "solution", of course, you mean finding $x,y,z \in \mathbb{N}$. Right? $\endgroup$ – user2468 Feb 27 '12 at 0:00
  • 1
    $\begingroup$ Note that it's enough to show that there are three rational numbers with product 1 and sum $q$. If we have $x/y = a/b$ and $y/z = c/d$ with $a,b,c,d$ integers, then $z/x = bd/ac$. We can then write $q = (ac)/(bc) + (bc)/(bd) + (bd)/(ac)$ to give a solution of the form you seek. $\endgroup$ – Michael Lugo Feb 27 '12 at 2:07
  • $\begingroup$ @Michael, I don't understand: given, say, $q=4$, how does your calculation find $x,y,z$ (or $a,b,c,d$) that work? $\endgroup$ – Gerry Myerson Feb 27 '12 at 2:33
  • $\begingroup$ It doesn't. But let's say, for example, that I had $q = 731/210$, and I had rational numbers whose product is 1, say $2/3, 5/7, 21/10$, which add up to $731/210$. Then I would be frustrated because the numerators and denominators don't match up. But here we have $a = 2, b = 3, c = 5, d = 7$ and so we can write $10/15 + 15/21 + 21/10$, which has the matching numerators and denominators. $\endgroup$ – Michael Lugo Feb 27 '12 at 3:54

The problem


with $N,x,y,z \in \mathbb{Z}$ was considered by Andrew Bremner and Richard Guy in "Two more representation problems" published in the Proceedings of the Edinburgh Mathematical Society, vol. 40 pp.1-17 in 1997. An online copy is available here. They showed solutions only occurred for those $N$ where the elliptic curve


has rank at least $1$.

For small $N>0$, the first solution is for $N=6$, with $x=18$, $y=4$ and $z=3$.

  • $\begingroup$ $N=5$ has a solution with $x=4, y=1, z=2$. +1 for the reference, though. $\endgroup$ – Aryabhata Feb 28 '12 at 18:49
  • 3
    $\begingroup$ Sorry!!!! N=5 is an outlier in the sense that its elliptic curve has rank 0 but torsion $\mathbb{Z}/6\mathbb{Z}$, whereas generally the torsion is $\mathbb{Z}/3\mathbb{Z}$. The order 6 torsion points give the quoted solution. $\endgroup$ – Allan MacLeod Feb 28 '12 at 19:45
  • $\begingroup$ Allan, thanks for clarifying! $\endgroup$ – Aryabhata Feb 28 '12 at 19:55
  • $\begingroup$ Thanks for answer , I'll study that paper though It seems beyond of my knowledge . Can someone find next $N$ with desired property ? (I'm not familiar with elliptic curve) $\endgroup$ – Mahan Feb 28 '12 at 22:11
  • $\begingroup$ Wonderful! It is another reason for me to try to learn something about elliptic curves. $\endgroup$ – user940 Mar 1 '12 at 20:05

Maple code that searches for solutions in specific range :

for q from 4 to 30 do
for x from 1 to 200 do
for y from 1 to 200 do
for z from 1 to 200 do
if x/y+y/z+z/x=q then
end if;
end do;
end do;
end do;
end do;

For $~q=9~$ ;$~(x,y,z)=(12,63,98)$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.