# Sequence of ratios of primes and integers

1. Mar 15, 2007

### Dragonfall

I am fairly certain that $$\frac{n}{p_n}$$ is not monotone for any n, but I can't give a proof of it without assuming something at least as strong as the twin prime conjecture. I was wondering if anyone has some advice to prove this using known methods?

2. Mar 15, 2007

### mathman

Statement is confusing - what is variable (not n from what you said)?

3. Mar 15, 2007

### Dragonfall

The variable is n. I mean no tail of the sequence is monotone.

4. Mar 20, 2007

### Dragonfall

Any ideas at all? I'm drawing dead here.

5. Mar 21, 2007

### CRGreathouse

I'm still not sure what you mean, exactly.

6. Mar 21, 2007

### Dragonfall

The tail of a sequence $$a_n$$ is the subsequence $$(a_n)_{n>N}$$ for some N. A sequence may not be monotone for the first N numbers, but the tail of the sequence might be monotone. If $$\frac{n}{p_n}$$ is eventually monotone, then the alternating series test shows that $$\sum_n(-1)^n\frac{n}{p_n}$$ converges. I believe that the series converges, but I don't think the sequence is eventually monotone, mainly because I think the twin prime conjecture is true.

Last edited: Mar 21, 2007
7. Mar 22, 2007

### Gib Z

I still have no idea what you mean, but $$\sum_{n=1}^{\infty} \frac{n}{p_n}$$ diverges if that helps.

8. Mar 22, 2007

### Dragonfall

Ok I don't know how I can possibly make it clearer. Look at this sequence:

2, 3, 5, 3, 5, 6, 7, 6, 4, 2, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, ...

This sequence is monotone decreasing from the 11th entry onwards. IS THE SAME TRUE FOR $$\frac{n}{p_n}$$? I don't think so, but it is not obvious either way.

9. Mar 22, 2007

### tehno

Just a quick suggestion

Have you tryed to link it with http://mathworld.wolfram.com/PrimeNumberTheorem.html" [Broken]?

Last edited by a moderator: May 2, 2017
10. Mar 22, 2007

### Dragonfall

All I can conclude from PNT is that $$\lim\frac{n}{p_n}=0$$.

11. Mar 22, 2007

### tehno

Dragonfall,I just realized what was your question in a first place...
I think it's darn difficult to prove that!
Need to know of distribution of primes for every finite segment [n,n+k] of natural numbers.Looks intractible at first glance.Sorry.

12. Mar 22, 2007

### DeadWolfe

No, it doesn't.

13. Mar 22, 2007

### Moo Of Doom

Why not? $\lim_{n\to \infty}\frac{n}{p_n}=0$ by the PNT, so wouldn't eventual monotonicity be sufficient for convergence?

Anyway, let's just see what we have.

No tail is monotone if for all natural numbers N, there is an n>N such that
$$\frac{n}{p_n} < \frac{n+1}{p_{n+1}}$$.

This is equivalent to
$$np_{n+1}<np_n+p_n$$
$$p_{n+1}-p_n<\frac{p_n}{n}$$

You don't need the twin primes conjecture here. It suffices to have this lemma:

There exists a natural number k such that there are infinitely many consecutive primes whose difference is less than k.

Even this is stronger than you need, but much, much weaker than the twin primes conjecture.

Last edited: Mar 22, 2007
14. Mar 22, 2007

### DeadWolfe

My deepest apologies, I somehow had forgotten the correct statement of the alternating series test.

Man do I feel stupid.

15. Mar 22, 2007

### Dragonfall

Thanks, and you're right, something like "there exists infinitely many n such that $$p_{n+1}-p_n<\log n$$" is approximately what I need.

16. Mar 23, 2007

### Moo Of Doom

Don't feel stupid. Stuff like that happens to me all the time.

Anyway, I did some looking, and I found this paper, which states that if the Elliot-Halberstam Conjecture is true, then there are infinitely many consecutive primes differing by less than 16.

I can't seem to find any similar results that don't depend on unproven conjectures, so proof might not be easy.

A proven result that might be useful is that ${\lim \inf}_{n \to \infty} \frac{p_{n+1}-p_n}{\log{p_n}}=0$ (found in the same paper).

17. Mar 23, 2007

### Dragonfall

I'm reading through it now, thanks. An immediate corollary is that $$\lim\inf\sqrt{p_{n+1}}-\sqrt{p_n}=0$$, which brings us infinitesimally closer to Andrica's conjecture (I haven't had time to think it through, it's just the first thing that popped into my head).

18. Mar 24, 2007

### Moo Of Doom

After a bit of tinkering and smoothing out some holes in the logic, I've come up with a proof of your conjecture using ${\lim \inf}_{n \to \infty} \frac{p_{n+1}-p_n}{\log{p_n}}=0$ as a lemma. I'll post it if you want to see.

19. Mar 24, 2007

### Dragonfall

I'll try and figure it out myself. If I get stuck maybe I'll crack and ask for your proof.

20. Mar 24, 2007

### Moo Of Doom

Fair enough. That's why I asked.

If you do manage to prove it, I'd love to see your proof. Please do post it.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook