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I understand that there are infinitely many primes and (obviously) infinitely many integers, but is there any way to calculate the total percentage of integers that are primes? Thanks

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    $\begingroup$ If you ask it this way, the correct answer is $0\%$, but you might be interested in a related question. Therefore, look at the prime number theorem $\endgroup$
    – Stefan Mesken
    Aug 13, 2014 at 7:00
  • $\begingroup$ Also questions 539192, 28540, 804502, and maybe more. $\endgroup$
    – Gerry Myerson
    Aug 13, 2014 at 7:22

1 Answer 1

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The function $\pi(n)$ tells us the amount of primes lower or equal to $n$. It is known that $\pi(n) \sim \frac{n}{\ln n}$ so the proportion as $n$ grows is about:

$$\frac{n}{\ln n}·\frac{1}{n} = \frac{1}{\ln n}$$

This means that as $n$ grows, the proportion tends to zero.

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  • $\begingroup$ Isn't it $\pi(n) \sim \frac{n}{\ln(n)}$? (Prime Number Theorem) $\endgroup$
    – rehband
    Aug 13, 2014 at 7:04
  • $\begingroup$ True, my bad. I'll correct it. $\endgroup$
    – Darth Geek
    Aug 13, 2014 at 7:06
  • $\begingroup$ Does this have anything to do with the circle constant π? $\endgroup$
    – Someone
    Mar 16 at 15:56

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