How to prove that there exists a prime number between $n$ and $ n!$, for all $ n> 2$?
(Bertrand's postulate gives a much better bound, but this question is about obtaining a self-contained proof.)
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this community
Base case: $3<5<3!$
Induction Hypothesis: Suppose that $\exists p$ such that $p$ is prime and $(n-1)<p<(n-1)!$
Induction Step: We must now show that $\exists k$ such that $k$ is prime and $n<k<n!$.
So either $n=p$ or $n<p<(n-1)!<n!$ If the case is the latter one, we are done. So, suppose that $n=p$. Now, we consider $Q=\prod p_i$ where $p_i\leq p$ and $p_i$ is prime. Well, we actually consider $Q+1$. Notice that $p<Q+1<n!$ Furthermore, notice that no primes less than or equal to $p$ divide $Q+1$ (because of the left over 1). Hence, either $Q+1$ is prime or composite. If $Q+1$ is prime, we are done. If $Q+1$ is composite (and since no prime less than or equal to $p$ divides $Q+1$) there must exist some $q<t<Q+1<n!$ such that $t|(Q+1)$. Thus, we are finished by setting $k=Q+1$ or $k=t$.