Bertrand's Postulate
If 
, there is always at least one Prime between 
and 
. Equivalently, if 
, then there is always at leastone Prime between and 
. It was proved in 1850-51 by Chebyshev, and is therefore sometimes known asChebyshev's Theorem. An elegant proof was later given by Erdös. An extension of this result is that if 
, thenthere is a number containing a Prime divisor 
in the sequence , 
, ..., 
. (The case 
thencorresponds to Bertrand's postulate.) This was first proved by Sylvester, independently by Schur, and a simple proof was givenby Erdös.
A related problem is to find the least value of 
so that there exists at least one Prime between and 
for sufficiently large (Berndt 1994). The smallest known value is 
(Lou and Yao 1992).
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, p. 135, 1994. Erdös, P. ``Ramanujan and I.'' In Proceedings of the International Ramanujan Centenary Conference held at Anna University, Madras, Dec. 21, 1987. (Ed. K. Alladi). New York: Springer-Verlag, pp. 1-20, 1989. Lou, S. and Yau, Q. ``A Chebyshev's Type of Prime Number Theorem in a Short Interval (II).'' Hardy-Ramanujan J. 15, 1-33, 1992.
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