In 1999, P. Dusart showed that for x ≥ 3275, there exists at least a prime number in
the interval
(x, x(1 + 1/(2ln²x))]
and in 2003, O. Ramaré and Y. Saouter showed that for x ≥
10726905041 there exists at least a prime number in the interval
(x(1 − ∆⁻¹), x], 
in which
∆ = 28314000. In this note, we show that for x ≥ 1.17×10¹⁶³⁴, we can yield Ramaré-Saouter’s
result from Dusart’s result.