On The Prime Numbers In Intervals

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.