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Number Theory

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Distribution of Twin Primes, True/False

NUMTH-1MXJ@B

We say that a pair $p$, $p+2$ are twin primes if both are prime numbers. Denote by $P$ the set of all prime numbers. Denote by $T$ the set of all pair of twin primes.

Which of the following statements are true? Select ALL that apply.

A

$\sum\limits_{p\in P} \frac1p <\infty$.

B

$\sum\limits_{(p, p+2)\in T} \frac1p <\infty$.

C

The number of primes $p\leq x$ such that $(p,p+2)\in T$ is $O\left(\frac x{\log^2 x}\right)$.

D

There are infinitely many $p\in P$ such that $(p, p+2)\notin T$.