How many prime numbers in a given interval?

Question

Is there any algorithm or a technique to calculate how many prime numbers lie in a given closed interval [a1, an], knowing the values of a1 and an, with a1,an ∈ ℕ?

Example:

[2, 10] --> 4 prime numbers {2, 3, 5, 7}

[4, 12] --> 3 prime numbers {5, 7, 11}

You may found this helpful. Prime counting function – prog_SAHIL Aug 29 '18 at 13:59 — prog_SAHIL, Aug 29 '18 at 13:59

gammatester · Accepted Answer · 2018-08-29T14:05:47.173

2

Use the prime-counting function and get $\pi(a_n)-\pi(a_1-1)$

edited Aug 29 '18 at 14:05

answered Aug 29 '18 at 13:56

gammatester

19,147

1

Since $a_1$ is included in the interval, I think your formula should be $\pi(a_n)-\pi(a_{1}-1)$ – paw88789 Aug 29 '18 at 14:01
@paw88789: Thanks, corrected. – gammatester Aug 29 '18 at 14:06

score 2 · Answer 2 · answered Aug 29 '18 at 13:59

The prime number theorem gives an approximation to $\pi(n),$ the number of primes less than $n$. You can subtract the values at the start and end of an interval. Mathematica and other software can compute exact values for small values of $n$ and approximate values for larger values of $n$. The first approximation is $\pi(n) \sim \frac n{\log n}$ but a much better one is $$\pi(n)\sim \int_2^n \frac {dt}{\log t}$$

How many prime numbers in a given interval?

2 Answers2