  We prove that given $\lambda \in \mathbb{R}$ such that $0 < \lambda < 1$,
then $\pi(x + x^\lambda) - \pi(x) \sim \displaystyle
\frac{x^\lambda}{\log(x)}$. This solves a long-standing problem concerning the
existence of primes in short intervals. In particular, we give a positive
answer (for all sufficiently large number) to some old conjectures about prime
numbers, such as Legendre's conjecture about the existence of at least two
primes between two consecutive squares.
