Find integer solutions to$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=3$

Question

I have been looking at this problem that has no positive integer solutions, and I became curious regarding the similar following problem:

I am trying to find solutions to$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$, where $a,b,c \in \mathbb{N}$.

It is fairly clear that $a=b=c=1$ is a solutions, and thus, $a=b=c=d\in \mathbb{N}$ is a family of solutions where each variable is the same. (Not terribly interesting). I am curious if there are other solutions and how to find them. If not, how could one prove there exist none?

Hint: AM $\ge$ GM and what are the conditions for AM $=$ GM? — achille hui, Apr 27 '22 at 15:58
This well-know application of AM-GM has been proved here many times, e.g. see the linked dupe (and its links). — Bill Dubuque, May 05 '22 at 08:24

score 2 · Accepted Answer · answered Apr 27 '22 at 16:01

2

AM-GM gives

$$\frac{\frac{x}{y} + \frac{y}{z} + \frac{z}{x}}{3}\geq\sqrt[3]{\frac{x}{y}\cdot\frac{y}{z}\cdot\frac{z}{x}} = 1$$

Where equality is achieved when $\frac{x}{y} = \frac{y}{z} = \frac{z}{x}$. This gives $x = \frac{y^2}{z}$, and substituting gives $y^3 = z^3$ i.e. $y = z$. Therefore, the only solutions are $x = y = z = k$ where $k\in\mathbb{Z}$.

answered Apr 27 '22 at 16:01

Gareth Ma

3,725

Hey, it's Gareth again! – Clyde Kertzer Apr 27 '22 at 16:03
@ClydeKertzer Hey, do you know me/do I know you? hahaha – Gareth Ma Apr 27 '22 at 16:04
You helped on my question here: https://math.stackexchange.com/questions/4436298/find-all-primes-a-b-c-such-that-a2a-b2b-c2c-form-an-arithm – Clyde Kertzer Apr 27 '22 at 16:05
Ah, the arithmetic progression one. Nice – Gareth Ma Apr 27 '22 at 16:05
Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque May 05 '22 at 08:24

Find integer solutions to$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=3$

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