Find the largest positive integer $n$ of the form $n=p^{2\alpha}q^{2\beta}r^{2\gamma}$ for primes $p<q, r$ and positive integers $\alpha, \beta, \gamma$, such that $|r-pq|=1$ and $p^{2\alpha}-1, q^{2\beta}-1, r^{2\gamma}-1$ all divide $n$.
Source: Bulgarian Autumn Tournament 2022 11.3
Tags: number theory
Find the largest positive integer $n$ of the form $n=p^{2\alpha}q^{2\beta}r^{2\gamma}$ for primes $p<q, r$ and positive integers $\alpha, \beta, \gamma$, such that $|r-pq|=1$ and $p^{2\alpha}-1, q^{2\beta}-1, r^{2\gamma}-1$ all divide $n$.