Q2 is impossible. If you have a
product of “large” primes minus a product of “large” primes then
this equation will be of the form (odd – odd) = even. So you have
to start with the first primes, starting with 2.
But using an updated threesome of
summands I found algebraic identities:
(p+6)(p+8) then I found p=4402241 yields consecutive primes
p(p+12)+(p+6)+(p+34)=(p+4)(p+10) with p=3457 and 101107 and
many more values.
I found the identities with a
little bit of logic (primes are of the form 6n+1,6n-1) and searching
by hand through possibilities. Then I ran the computer to find the
specific gap sequence I wanted.
One might argue that a single
factor does not a product make.