What numerical example does the video use to check the prime density approximation at $x = 100$ trillion?

Q: What numerical example does the video use to check the prime density approximation at x = 100 trillion?

The video evaluates the density approximation 1/ln(x) at x = 100 trillion, yielding an estimated density of 3.1%. It compares this with the actual counted density of 3.2%, noting a 0.1% discrepancy that shrinks for larger numbers.

The video evaluates the density approximation

1/\ln (x)

x = 100

trillion, yielding an estimated density of 3.1%. It compares this with the actual counted density of 3.2%, noting a 0.1% discrepancy that shrinks for larger numbers.

Conditions

$x = 100$ ,000,000,000,000.
Density approximation is #PRIMES/x ~ $1/\ln (x)$ .

Reasoning, step by step

Substitute $x = 100$ trillion into $1/\ln (x)$ .
Evaluate ln(100 trillion) as approximately 32.23.
Calculate the reciprocal $1/32.23$ to get 3.1%.
Compare the estimate 3.1% with the actual density 3.2%.

Example

The screen displays $x = 100000000000000$ , then $1/32.23$ , then 3.1%, and finally 3.2% ~ 3.1%.

Common misconceptions

Assuming the 0.1% difference means the formula is invalid; the video states the difference approaches zero for larger numbers.
Confusing the density percentage with the absolute count of primes.

Watch the explanation

YouTubeThe Prime Number Theorem

4:46 – 5:08Watch this moment ↗

Explore next

Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.

What numerical example does the video use to check the prime density approximation at x=100x = 100x=100 trillion?