Semiprime Antics

[Chuck Seggelin]

Hi there,

I was working on some new sequences last night involving semiprimes and I
encountered this curiousity.  Take a semiprime N (a number which is the
product of two primes, see A001358) and its two prime factors J and K and
test the following rules:

    r1: N+J-K is prime
    r2: N-J+K is prime
    r3: N+J+K is prime
    r4: N-J-K is prime

What I noticed was that for any combination of three (or less) of these
rules, there are abundant semiprimes that pass the test.  But if one tries
to enforce all four rules, there appear to be only two: 10 and 15.

I've currently tested semiprime terms up to 97,232,493 and haven't found any
further terms that satisfy all four rules.  I'm beginning to wonder if I'm
chasing after something which is mathematically impossible for any
semiprimes other than 10 and 15.  I'm just an amateur so it is highly
possible that I am missing something.

Is there an obvious reason why no semprimes other than 10 and 15 can adhere
to all four rules above?

Thanks!

-- Chuck Seggelin