On the distribution of "nice" primes (primes $p$ , such that $\pi(p)$ is prime as well)

"nice" primes (primes $p$, such that $\pi(p)$ is prime as well) are introduced.

Is it known whether the sum of the reciprocals of the nice primes is convergent ?

The sum of the reciprocals of the primes in an arithmetic progression $an+b$ , where $a$ and $b$ are coprime and $n$ runs over the natural numbers, is divergent.

Intuitively, I would expect that there should exist enough nice primes to get a divergent series, but I may be wrong.

Has anyone an idea or a reference ?

  • $\begingroup$ Another thing I am interested in : What is the largest known "nice" prime ? $\endgroup$ – Peter Jul 29 '16 at 23:18

Because of the Prime Number Theorem, ultimately we have $$.9\,n\,\log n \leq p_n \leq 1.1 \,n\,\log n$$ where $p_n$ is the $n$th prime.

The $n$th nice prime is $p_{p_n}$, so we ultimately get $$p_{p_n} \geq .9\,p_n\,\log(p_n)\geq .9 \,(.9\,n\,\log n)\,\log(.9\,n\,\log n).$$

Since that last logarithm is equivalent to $\log n$, we ultimately get $$p_{p_n} \geq .5\,n\,(\log n)^2$$ so $$\frac 1{p_{p_n}} \leq \frac{2}{n\,(\log n)^2} \qquad\text{ultimately,}$$ which proves that $\sum_n \frac{1}{p_{p_n}}$ converges, because $\sum \frac 1{n(\log n)^\alpha}$ converges iff $\alpha > 1$ (that's a Bertrand series, it's for example an application of Cauchy's condensation test or of an integral comparison).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.