Here m is in twin prime pair:(6m-1, 6m+1) and n is nth twin prime-pair. I am just interested to know lower bound of difference of consecutive first twin primes as n tends to infinity. For example in (5, 7) and (11, 13): Consecutive first twin primes means 5 and 11. I know that sum of reciprocal of twin primes converges due to Brun' s Constant. So, the sum of reciprocals of first twin primes converges. I can use Raabe's test which is sufficient to test for convergence. On applying it I can get the lower bound I am talking about.
2026-03-26 21:35:17.1774560917
What is m/n as n tends to infinity?
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It is conjectured that there are infinitely many integers $n$ such that $n$, $n+2$, $n+6$, and $n+8$ are simultaneously prime (since $\{0,2,6,8\}$ is an "admissible tuple" for the Hardy–Littlewood conjectures). Therefore the best possible lower bound for the difference between consecutive twin primes is $6$. (I'm answering the question in the body of the post—the title does not match it.)