1. If p is prime and p+k is prime then p+k divides p^(p+k) + k.
2. the converse
If p is prime and p+k is composite and p+k divides p^(p+k) +k, then p+k is a
For k=2 we get the case for twin primes. There are 33 pseudotwinprimes for
primes less than 130000. In other words of the 12159 primes p less than
130000 the divide
by p+2 test failed 33 times to detect if p and p+2 were or were not twin
for k=4 or quad primes, there are 11 pseudoquadprimes for 344 quadprimes
less than 20000.
Thus it appears, at least statistically, that it is highly probable that
p+k is prime if p+k divides
p^(p+k) + k.
It is interesting to note that Charmichael numbers pop up in these
Maybe someone can prove 1. and elaborate. I have included a pari script
pikprimes(n,k) = \\The number of k difference prime pairs less than n.