Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I've been thinking about primorials in the context of the twin prime conjecture. I am seeing this primarily as an exercise to improve my intuition about primorials and prime patterns more than the possibility of making any progress. :-)

So, I thought that it would be interesting to think about numbers of the following form. Let $p_k$ be the $k$th prime. Then, I am interested in the following sequence of numbers:

$\left(p_k\#-1, p_k\#+1\right)$, $\left(\frac{p_k\#}{3}+1,\frac{p_k\#}{3}+3\right)$, $\left(\frac{p_k\#}{3}-1,\frac{p_k\#}{3}-3\right)$, $\left(\frac{p_k\#}{2}+2,\frac{p_k\#}{2}+4\right)$, $\left(\frac{p_k\#}{2}-2,\frac{p_k\#}{2}-4\right)$, $\left(\frac{p_k\#}{15}+3,\frac{p_k\#}{15}+5\right)$, $\left(\frac{p_k\#}{15}-3,\frac{p_k\#}{15}-5\right) \cdots$

Here is my question. Does anyone have any suggestions on methods for counting the number of distinct integeres in this sequence? My basic goal here is to estimate a lower bound for the number of distinct $x$ where $p_k< x \le p_k\#+1$

share|cite|improve this question
I'm not clear on the definition of the sequence, could you be more specific? – Charles Jul 25 '13 at 14:56
The sequence is an effort is to use the primorial to create a sequence of possible twin prime pairs. So the first value is (-1,+1), the next one is (+1,+3) and (-1,-3), the one after that is (+2,+4) and (-2,-4) etc. I hope that explains the sequence. For my purposes, I would be fine if you provide a lower estimate of just the positive values of the sequence. – Larry Freeman Jul 26 '13 at 2:40
You're still not explaining where -1, +1, +1, +3, -1, -3, +2, +4, -2, -4, etc. come from. – Charles Jul 26 '13 at 13:46
Thanks for your question. The sequence is just counting. I am going 1,2,3,4,5,6 ,etc. then I am adding the sequence number to a primorial divided by the common factors between the sequence number and the primorial. So, assume we have a form of $\frac{p_k}{a} + c,\frac{p_k}{a}+c+2$ where $a | p_k\#$ and $a | c(c+2)$. Does that answer explain where I am going? – Larry Freeman Jul 26 '13 at 22:30
up vote 1 down vote accepted

You asked for a bound on the number of integers between $p_k$ and $p_k\#+1$. $\log x\#\sim x$ and so the number is roughly $e^{p_k}\approx k^k.$

share|cite|improve this answer
Wow. Thanks very much for the answer! I would be very interested if you could provide details explaining your results. – Larry Freeman Jul 26 '13 at 2:38
To be clear, I really like your analysis for showing that $p_k$# is approximately $k^k$. I am interested in understanding how this applies the number of distinct integers in the sequence that I describe. Are you saying the lower bound of the sequence is $k^k$. If so, I would be very interested in understanding your reasoning. – Larry Freeman Jul 26 '13 at 3:50
It's approximate, not a lower bound. If you replace e with any smaller number it becomes a lower bound for all but finitely many numbers. The prime number theorem says that x# is about exp(x) and p_k is about k log k; combining the two gives the result. – Charles Jul 26 '13 at 13:45

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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