Question about density of prime numbers?

[goldust]

It is known that prime numbers become sparser and sparser, with the average distance between one prime number and the next increasing as n approaches infinity. Dividing an even number by 2 results in a bottom half from 1 to n / 2 and a top half from n / 2 to n. For a particular sufficiently large n, would the top half from n / 2 to n no longer contains any prime numbers?

 

[eigenperson]

No, there is always a prime between n and 2n (which is the same thing as between n/2 and n). This is called Bertrand's Postulate, but despite the name, it's a theorem.

In fact, there are a LOT of primes in that interval, and the bigger n is, the more there are. The density of the primes near n is like ${1 \over \log n}$, so a rough estimate for the number of primes between n and 2n is ${n \over \log 2n} \approx {n \over \log n}$.