Ivan Ianakiev sent the following nice conjecture
"Let p denote any prime in any decomposition of 2n. 4 and 6
are the only numbers n such that 2n + p is prime for every
8=3+5; then 8+3 & 8+5 are primes.
12=5+7; then 12+5 &12+7 are primes.
Can you find another example or prove the conjecture?