MathNet 2009-part-02 — 175 problems (EN–VI) Question 125

Let $a$, $b$, $c \gt 1$ be positive integer numbers. Prove that there exist infinitely many $a$, $b$, $c$ such that $a|b^2 - 1$, $b|c^2 - 1$, $c|a^2 - 1$.
(proposed by B. Battsengel)