Proof of convergence
Everywhere below will stand for an odd prime number. Let . We shall prove that for large with some absolute constant . The technique used in the proof is a version of the inclusion-exclusion principle and is known nowadays as Brun's simple pure sieve.
Let . Let be the -th symmetric sum of the numbers . Then for every odd and even .
Proof of Lemma
Induction on .
Now take a very big and fix some to be chosen later. For each odd prime let
Clearly, if , and for some , then either or is not prime. Thus, the number of primes such that is also prime does not exceed .
Let now be an even number. By the inclusion-exclusion principle,
Let us now estimate . Note that the condition depends only on the remainder of modulo and that, by the Chinese Remainder Theorem, there are exactly remainders that satisfy this condition (for each , we must have or and the remainders for different can be chosen independently). Therefore
where . It follows that
where is the -th symmetric sum of the set . Indeed, we have not more than terms in the inclusion-exclusion formula above and each term is estimated with an error not greater than .
Now notice that by the lemma. The product does not exceed (see the prime number article), so it remains to estimate . But we have
This estimate yields the final inequality
It remains to minimize the right hand side over all possible choices of and . We shall choose and . With this choice, every term on the right does not exceed and we are done.