Primes in [kn, (k + 1)n] , for 1⩽. k ⩽. 10

Abstract : In 1919 S. Ramanujan proved [Ram] the Bertrand postulate elegantly with approximations based on the Stirling formula. P. Erdös proved also [Erd] the same result in 1931 by a carefull study of the binomial coeficient (2n n). The Bertrand postulate was first proved by P. Tchebichef in 1850. Without the use of the PNT and returning to the inequalities of Tchebichef, we have a fast proof for existence of primes in the interval [kn, (k + 1)n] for k ∈ {1, 2, 3, 4}. Using the improvements of J.J. Sylvester (1892) [Syl] the result is given for 1 k 10. Soit θ(x) la somme des logarithmes de tous les nombres premiers ne dépassant pas x et soit
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Tarek Zougari. Primes in [kn, (k + 1)n] , for 1⩽. k ⩽. 10. 2016. ⟨hal-01303279⟩

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