# Is it true/known/important that $(\log p_n)/n$ is nonincreasing, where $p_n$ is the $n$th odd prime number?

First thing first, I would like to apologize in advance for my poor knowledge of Maths and English. I'm an Italian student and after asking to all the mathematicians and Maths teachers in my town, I hope that the internet might help me regarding the following conjecture.

Conjecture: If $p_{n}$ is the $n$th odd prime, then $$\frac{\log_{10}p_{n}}{n} \geq \frac{\log_{10}p_{n+1}}{n+1}.$$

To date, I've tested it for the first 64k prime numbers.

My questions are:

-Does it deserve to be shown to other people/tried to prove, or is it just "one among thousands"?

-If it's important, who could I show it to?

-If it isn't important, what makes some conjectures noteworthy?

• I hope the translation shows I decipher your conjecture right. :) No apology is necessary. – Megadeth Nov 6 '15 at 7:17
• Moreover, as $\log_{10}x = \frac{\log x}{\log 10}$, you can simply write $$\frac{\log p_{n}}{n} \geq \frac{\log p_{n+1}}{n+1}.$$ – Megadeth Nov 6 '15 at 7:18
• Hint: 0.92929 pn / ln pn<= n<= 1.2556 pn / ln pn , log pn = ln pn / ln 10 – some one Nov 6 '15 at 7:20
• See the following paper arxiv.org/pdf/1208.2683 – Kelenner Nov 6 '15 at 7:33