The semiprime $87 = 3*29$ has a curious property: it's the fact that both
$87^2 + 29^2 + 3^2 = 8419$
and
$87^2 - 29^2 - 3^2 = 6719$
are prime numbers.
This intrigued me and led me to wonder if there are other semiprimes with the same property, and I found that
$21 = 3*7$ is another example, since both
$21^2 + 7^2 + 3^2= 499$
and
$21^2 - 7^2 - 3^2 = 383$ are prime numbers
So the following question arises: Are there infinitely many prime numbers $p$ and $q$, with $p \neq q$, such that both
$(pq)^2 + p^2 + q^2$
$(pq)^2 - p^2 - q^2$
are also primes?
Does this follows from some known theorem or conjecture?