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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
##### Citations:
Zbl 0004.10103; Zbl 1154.11300; JFM 58.0154.04
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