Diophantus of Alexandria/Arithmetica/Book 1/Problem 22

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Example of Diophantine Problem

To find $3$ numbers such that, if each give to the next following a given fraction of itself, in order, the results after each has given and taken may be equal.

That is:

Let $\dfrac 1 p, \dfrac 1 q, \dfrac 1 r$ be given.

The exercise is to find a set of $3$ natural numbers $\set {x, y, z}$ such that:

\(\ds x - \frac x p + \frac z r\) \(=\) \(\ds m\)
\(\ds y - \frac y q + \frac x p\) \(=\) \(\ds m\)
\(\ds z - \frac z r + \frac y q\) \(=\) \(\ds m\)

where $m$ is a natural number.

Example: $\dfrac 1 3, \dfrac 1 4, \dfrac 1 5$


the first give $\dfrac 1 3$ of itself to the second
the second give $\dfrac 1 4$ of itself to the third
the third give $\dfrac 1 5$ of itself to the first.

What are the numbers?