Rosser's theorem

This article is about a theorem in number theory. For Rosser's technique for proving incompleteness theorems, see Rosser's trick. For Gödel–Rosser incompleteness theorems, see Gödel's incompleteness theorems. For the Church-Rosser theorem of λ-calculus, see Church-Rosser theorem.

In number theory, Rosser's theorem was proved by J. Barkley Rosser in 1938. Its statement follows.

Let pn be the nth prime number. Then for n ≥ 1

${\displaystyle p_{n}>n\cdot \ln n.}$

This result was subsequently improved upon to be:

${\displaystyle p_{n}>n\cdot (\ln n+\ln(\ln n)-1).}$ (Havil 2003)