Approximate Value of Nth Prime Number
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Theorem
The $n$th prime number is approximately $n \ln n$.
Proof
This will be demonstrated by showing that:
- $\displaystyle \lim_{n \mathop \to \infty} \dfrac {p_n} {n \ln n} = 1$
where $p_n$ denotes the $n$th prime number.
By definition of prime-counting function:
- $\map \pi {p_n} = n$
The Prime Number Theorem gives:
- $\displaystyle \lim_{x \mathop \to \infty} \dfrac {\map \pi x} {x / \ln x} = 1$
Thus:
- $\displaystyle \lim_{x \mathop \to \infty} \dfrac n {p_n / \ln p_n} = 1$
| \(\displaystyle \lim_{x \mathop \to \infty} \dfrac {\map \pi x} {x / \ln x}\) | \(=\) | \(\displaystyle 1\) | Prime Number Theorem | ||||||||||
| \(\displaystyle \leadsto \ \ \) | \(\displaystyle \lim_{n \mathop \to \infty} \dfrac n {p_n / \ln p_n}\) | \(=\) | \(\displaystyle 1\) | Definition of Prime-Counting Function | |||||||||
| \(\text {(1)}: \quad\) | \(\displaystyle \leadsto \ \ \) | \(\displaystyle \lim_{n \mathop \to \infty} \dfrac {p_n} {n \ln p_n}\) | \(=\) | \(\displaystyle 1\) | |||||||||
| \(\displaystyle \leadsto \ \ \) | \(\displaystyle \lim_{n \mathop \to \infty} \paren {\ln p_n - \ln n - \ln \ln p_n}\) | \(=\) | \(\displaystyle 0\) | Logarithm of both sides | |||||||||
| \(\displaystyle \leadsto \ \ \) | \(\displaystyle \lim_{n \mathop \to \infty} \ln p_n \paren {1 - \frac {\ln n} {\ln p_n} - \frac {\ln \ln p_n} {\ln p_n} }\) | \(=\) | \(\displaystyle 0\) | multiplying argument through by $\ln p_n$ | |||||||||
| \(\displaystyle \leadsto \ \ \) | \(\displaystyle \lim_{n \mathop \to \infty} \ln p_n \paren {1 - \frac {\ln n} {\ln p_n} }\) | \(=\) | \(\displaystyle 0\) | as $\dfrac {\ln n} n \to 0$ | |||||||||
| \(\displaystyle \leadsto \ \ \) | \(\displaystyle \lim_{n \mathop \to \infty} \paren {1 - \frac {\ln n} {\ln p_n} }\) | \(=\) | \(\displaystyle 0\) | as $\ln p_n \ne 0$ | |||||||||
| \(\displaystyle \leadsto \ \ \) | \(\displaystyle \lim_{n \mathop \to \infty} \frac {\ln n} {\ln p_n}\) | \(=\) | \(\displaystyle 1\) | as $\ln p_n \ne 0$ | |||||||||
| \(\displaystyle \leadsto \ \ \) | \(\displaystyle \lim_{n \mathop \to \infty} \dfrac {p_n} {n \ln n}\) | \(=\) | \(\displaystyle \lim_{n \mathop \to \infty} \dfrac {p_n} {n \ln p_n} \frac {\ln p_n} {\ln n}\) | from $(1)$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle 1\) |
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.16$: The Sequence of Primes
