Cramér's prime gap conjecture

Statement

This conjecture states that the prime gap between a prime $p$ and the next prime is $O((\log p{)}^2)$. In other words, there exists a constant $c$ such that, for any prime $p$, there exists a prime $q$ such that $p<q<p+c(\log p{)}^2$.

Equivalently, for any natural number $n>1$, there exists a prime $q$ such that $n<q<n+c(\log n{)}^2$.

Relation with other facts and conjectures

Weaker facts and conjectures

• Bertrand's postulate
• The generalized Riemann hypothesis implies a prime gap of $O(\sqrt{n}\log n)$.
• The prime-between-squares conjecture states that there is a prime between the squares of any two consecutive natural numbers.