We prove that if x is large enough, namely x≥ x₀, then there exists a prime between x(1- Δ⁻¹) and x, where $Δ$ is an effective constant computed in terms of x₀. This improves some previous results of Ramaré and Saouter and is used to verify the Ternary Goldbach Conjecture for the integers up to 7.86· 10²⁷.
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Kadiri et al. (2014) studied this question.