19.3.13 Find a set of prime numbers \( p, q, r \), such that the numbers \( p(p+1), q(q+1), r(r+1) \) form an increasing arithmetic sequence.
According to the identity
$$
n(n+1)+(41 n+20)(41 n+21)=2(29 n+14)(29 n+15),
$$

the numbers $n(n+1),(29 n+14)(29 n+15)$, and $(41 n+20)(41 n+21)$ form an arithmetic sequence. If for some positive integer $n$, the numbers $n, 29 n+14, 41 n+20$ are all prime, then they constitute a solution to this problem. The smallest such $n=127, p=127, q=3697, r=5527$ is a solution to this problem.