How accurate is the $x/\ln (x)$ approximation for the number of primes below 100 trillion?

Conditions

• $x = 100$,000,000,000,000.
• The formula used is #primes(x) ~ $x/\ln (x)$.

Reasoning, step by step

1. Substitute $x = 100$ trillion into $x/\ln (x)$.
2. Evaluate the expression to get approximately 3.1 trillion.
3. Compare this estimate with the actual computationally verified count of 3.2 trillion.
4. Calculate the accuracy percentage.

Example

The video states 100 trillion divided by the natural log of 100 trillion equals 3.1 trillion, and compares it to the actual count of 3.2 trillion, noting it is 99.996875% accurate.

Common misconceptions

• Assuming the approximation is only useful for theoretically infinite x; the video shows it is highly accurate even at 100 trillion.
• Confusing the density approximation (3.1%) with the count approximation (3.1 trillion).

Watch the explanation

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