Why does the following simple expression $p\cdot q +(p+q)$ generate so many primes? We give a simple example:  
$$3\cdot5 + (3+5) = 23$$
$$3\cdot7 + (3+7) = 31$$
$$3\cdot11 + (3+11) = 47$$
The expression also works if we use $pN + (p+N)$ with $N=rs$, $r,s$ primes.
Is there a simple explanation?
 A: This is similar to Conjecture 7.1 of this paper by Carl Pomerance and Simon Rubinstein Salzedo. 
Their conjecture 7.1 follows from the statement that, for infinitely many primes $p$, there exists a prime $q>p$ such that $pq-p-q$ is also prime. This statement follows from Dickson's prime $k$-tuples conjecture. For more explanation, for fixed $p$, the two linear polynomials $x$ and $(p-1)x-p$, can be simultaneously prime for infinitely many $x$. 
But, Dickson's prime $k$-tuples conjecture is open. 
For this problem, fix an odd prime $p$, consider the two linear polynomials $x$ and $(p+1)x+p$. From Dickson's conjecture, these are both primes for infinitely many $x$. 
Combining with Hardy Littlewood conjecture (also open), we have the following conjectural formula for a fixed odd prime $p$.
$$
\#\{q<X \  | \ q, (p+1)q+p \textrm{ are both primes}\} \gg \frac{X}{\log^2 X}. 
$$
A: Well, if $p$ and $q$ are odd primes it's not divisible by $2$ or $p$ or $q$.  For any other prime $r$, if $q\equiv -1$ mod $r$ it won't be divisible by $r$, while otherwise it's divisible by $r$ only if $p \equiv -q (1+q)^{-1} \mod r$, which for "random" prime $p$ has probability $1/(p-1)$.
A: Part of the reason, is only one combination type or residues mod 6, that represent at least 2 primes, give back numbers that aren't in prime residues. The  exceptions, are when both $p,q$ are  both 1,2,3 mod 6. I included 2 and 3 in my checking.
Small examples are only great as counter examples of something holding generally. $13\times 7+(13+7)= 111=3\times 37$ for example. 
of the 10 distinct cases, 7 produce prime residues that just don't produce one prime in the integers, 6 of which produce the same residue ( 5 mod 6 is higher). 
Your examples are also kind of low just checking p,q both 5 mod 6 up to 521, shows only about 31% of combinations actually produce primes. 
Similar results with the 1,5 combination.  about 41% in the p=3 case.
and finally roughly 47% in the p=2 case. 
Because of weighting that means ... under 18% of  possibly prime cases are primes.(894/5092)
and 9604 cases total if you include those that can't work. which brings us under 10% 
