When I look up the prime number article in wikipedia, I find this theorem interesting. It is related about the attempt for mathematicians to determine the distribution of prime numbers. This theorem was first conjectured by Bertrand in 1845, so it is sometimes called Bertrand's postulate. The theorem states that for every integer $n>1$, there always exists at least one prime number $p$ where $n < p < 2n$. Bertrand verified this statement for all numbers up to 3 millions by himself (wow!). Then, five years later, it was proved by Chebyshev. However, it used non-elementary and it is a complex proof.

Later, the simpler proof was given by Paul Erdos in 1932. He also extend the Chebyshev's theorem to a more general ones, which states that for any positive integer $k$, there is a natural number $N$ such that for all $n > N$, there are at least $k$ primes between $n$ and $2n$.

There is another similar conjecture made by Legendre which says that there exists a prime between $n^2$ and $(n+1)^2$. However, it is still unproven so far.

Chebyshev's Theorem