Determine all triples $(m,n,p)$ of positive rational numbers.

Question

Determine all triples $(m,n,p)$ of positive rational numbers such that the numbers $m+\frac{1}{np}, n+\frac{1}{pm}, p+\frac{1}{mn}$ are integers.

I have no idea how to go about. Please help.

Answer 1

First, by multiplying, we can see that $\frac{(mnp+1)^3}{(mnp)^2}$ is an integer. Consider $mnp = \frac{X}{Y}$, where $\gcd(X,Y)=1$. Then, you can see that $X=Y=1$. This gives you $mnp=1$. Replace $m=\frac{w}{x}$, $n=\frac{y}{z}$, $p=\frac{xz}{wy}$, we can see using the three statements with divisibility solving that the only solutions are $(1,1,1)$ , $(2,1,\frac{1}{2})$ and $(4,\frac{1}{2},\frac{1}{2})$, and all its symmetric permutations.

The statements we get are $x \mid 2w$, $z \mid 2y$ and $wy \mid 2xz$. You can take casework to solve these.