Putnam

Only the primes 2 and 5 appear seven or more times. The fact that these primes appear is demonstrated by the examples $$(2,5,2), (2, 5, 3), (2, 7, 5), (2, 11, 5)$$ and their reversals. It remains to show that if either $\ell=3$ or $\ell$ is a prime greater than 5, then $\ell$ occurs at most six times as an element of a triple in $S$. Note that $(p,q,r) \in S$ if and only if $q^2 - 4pr = a^2$ for some integer $a$; in particular, since $4pr \geq 16$, this forces $q \geq 5$. In particular, $q$ is odd, as then is $a$, and so $q^2 \equiv a^2 \equiv 1 \pmod{8}$; consequently, one of $p,r$ must equal 2. If $r=2$, then $8p = q^2-a^2 = (q+a)(q-a)$; since both factors are of the same sign and their sum is the positive number $2q$, both factors are positive. Since they are also both even, we have $q+a \in \{2, 4, 2p, 4p\}$ and so $q \in \{2p+1, p+2\}$. Similarly, if $p=2$, then $q \in \{2r+1, r+2\}$. Consequently, $\ell$ occurs at most twice as many times as there are prime numbers in the list $$2\ell+1, \ell+2, \frac{\ell-1}{2}, \ell-2.$$ For $\ell = 3$,$\ell-2= 1$ is not prime. For $\ell \geq 7$, the numbers $\ell-2, \ell, \ell+2$ cannot all be prime, since one of them is always a nontrivial multiple of 3.