By asking this question, I hope that someone, somewhere, has an answer, not to the conjecture, but the question itself. Then, that person has proved an unsolved conjecture. I do have a proof, which is quite long, that all Germain primes are of the form 6n-1. For more details, go to the question: Are there an infinity of Sophie Germain primes? My second answer provides a full proof.
2026-04-01 03:00:12.1775012412
Could somebody prove that there are infinite values of a and b where both 6a-1 and 12b-1 are prime?
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$2(6n-1)+1=12n-1$. Thus, if $6n-1$ and $12n-1$ are both prime, then $6n-1$ is a Sophie Germain prime and $12n-1$ is the associated safe prime. Thus, for both to be prime infinitely often, there would have to be infinitely many Sophie Germain primes. As this is currently an unproved conjecture (at least according to Wikipedia), you probably won't get a proof here.