3

The sum $\sum_{k=1}^n1/p_k$, where $p_k$ is the $k$th prime number, is known to diverge as $O(\ln\ln n)$. However, prompted by Euclid, suppose that we restrict our choice of primes to a subset $Q:=\{q_1, q_2,...\}\subset\{p_1,p_2,...\}$, where $q_0:=1$ and $q_{n+1}$ is the least prime divisor of $q_0\cdots q_n+1$ (so that the first few elements of $Q$ are $2,3,7,43,13,53,\dots\,$). The question is: $$\text{does }\,\sum_{q\in Q}\frac1q\; \text{ exist?}$$

John Bentin
  • 20,004
  • Considering that even the sum of the reciprocals of the primes of the progression $an+b$ , where $a$ and $b$ are coprime positive integers, diverges, my guess is that this sum still diverges. It could even contain every prime number. I do not think that we can decide this interesting question ! – Peter Jan 23 '22 at 14:15
  • 3
    The sequence is called the Euclid-Mullin sequence. Shanks conjectured that it contains all the prime numbers. – jjagmath Jan 23 '22 at 14:18
  • @jjagmath A reasonable conjecture since the smallest not yet occured prime number $p$ should have a chance of $\frac{1}{p}$ to occur in the next step. – Peter Jan 23 '22 at 14:19
  • See also for example this post, or this one. – Dietrich Burde Jan 23 '22 at 14:28
  • This is a very interesting problem, I am very interested in this sequence of primes. Please post an update here in the future if you ever have one. – C Bagshaw Jan 26 '22 at 11:02

0 Answers0