2023/07/17 by Ferreira, Luan Alberto
#FOS: Mathematics #Number Theory (math.NT) #Primary 11N05 #Secondary 11L20
paper · doi:10.48550/arxiv.2307.08725
We prove that given λ∈ ℝ such that 0 < λ< 1, then π(x + xλ) - π(x) ∼ (xλ)/(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.