Bertrand's postulate

Statement

Let $n$ be a natural number greater than $1$. Then, there exists a prime number $p$ such that $n<p<2n$.

In other words, the prime gap, i.e., the gap between a prime $p$ and the next prime, is strictly smaller than $p$.

Relation with other facts and conjectures

Upper bounds on the limit superior of prime gap

• Cramér's prime gap conjecture states that the prime gap (i.e., the gap between a prime and the next prime) is $O((\log p{)}^2)$.
• The prime-between-squares conjecture states that there exists a prime between the squares of any two consecutive natural numbers.
• The Riemann hypothesis implies the large prime gap conjecture, which states that the prime gap is $O(\sqrt{p}\log (p))$.

Lower bounds on the limit superior of prime gaps

• Rankin's bound states that there exist arbitrarily large primes $p$ for which the prime gap is:

$\mathrm{\Omega }(\log p\frac{\log \log p\log \log \log \log p}{(\log \log \log p{)}^2}$.