  Bertrand's postulate establishes that for all positive integers $n>1$ there
exists a prime number between $n$ and $2n$. We consider a generalization of
this theorem as: for integers $n\geq k\geq 2$ is there a prime number between
$kn$ and $(k+1)n$? We use elementary methods of binomial coefficients and the
Chebyshev functions to establish the cases for $2\leq k\leq 8$. We then move to
an analytic number theory approach to show that there is a prime number in the
interval $(kn, (k+1)n)$ for at least $n\geq k$ and $2\leq k\leq 519$.
  We then consider Legendre's conjecture on the existence of a prime number
between $n^2$ and $ (n+1)^2$ for all integers $n\geq 1$. To this end, we show
that there is always a prime number between $n^2$ and $(n+1)^{2.000001}$ for
all $n\geq 1$. Furthermore, we note that there exists a prime number in the
interval $[n^2,(n+1)^{2+\varepsilon}]$ for any $\varepsilon>0$ and $n$
sufficiently large.
  We also consider the question of how many prime numbers there are between $n$
and $kn$ for positive integers $k$ and $n$ for each of our results and in the
general case. Furthermore, we show that the number of prime numbers in the
interval $(n,kn)$ is increasing and that there are at least $k-1$ prime numbers
in $(n,kn)$ for $n\geq k\geq 2$.
  Finally, we compare our results to the prime number theorem and obtain
explicit lower bounds for the number of prime numbers in each of our results.
