Why does the cumulative number of primes below x increase while prime density decreases?

Conditions

• x increases through the displayed powers of ten.
• Density is defined as (# primes)/(size).

Reasoning, step by step

1. Observe the table of prime counts for increasing SIZE values.
2. Note that the absolute count of primes strictly increases.
3. Compute the density ratio for the same values.
4. Observe that the percentage of primes decreases as SIZE grows.

Example

The video shows the count rising from 25 to 1229 as SIZE goes from 100 to 10000, but the density drops from 25% to 12.29%.

Common misconceptions

• Assuming that because more primes are found in larger ranges, primes must also become more common per integer.
• Reading a slowly rising cumulative count curve as indicating a constant or increasing density.

Watch the explanation

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