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Primes in the interval \([2n,3n]\). (English) Zbl 1154.11303
Summary: Is it true that for all integers \(n>1\) and \(k\leq n\) there exists a prime number in the interval \([kn, (k+1)n]\)? The case \(k=1\) is Bertrand’s postulate which was proved for the first time by P. L. Chebyshev in 1850, and simplified later by P. Erdős in 1932 [Acta Litt. Sci. Szeged 5, 194–198 (1932; Zbl 0004.10103 and JFM 58.0154.04)]. The present paper deals with the case \(k=2\). A positive answer to the problem for any \(k\leq n\) implies a positive answer to the old problem whether there is always a prime in the interval \([n^2,n^2+n]\) [see T. M. Apostol, Introduction to analytic number theory, Springer, New York (1998; Zbl 1154.11300), p. 11)].

MSC:
11A41 Primes
11N05 Distribution of primes
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