Difference between revisions of "Bertrand postulate"
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| − | For any natural number $n>3$ there exists a [[Prime number|prime number]] that is larger than $n$ and smaller than $2n-2$. In its weaker formulation Bertrand's postulate states that for any $x>1$ there exists a prime number in the interval $(x,2x)$. The postulate was advanced by J. Bertrand in 1845 on the strength of tabulated data, and was proved by P.L. Chebyshev (cf. [[Chebyshev theorems on prime numbers|Chebyshev theorems on prime numbers]]). | + | For any natural number $n>3$ there exists a [[Prime number|prime number]] that is larger than $n$ and smaller than $2n-2$. In its weaker formulation Bertrand's postulate states that for any $x>1$ there exists a prime number in the interval $(x,2x)$. The postulate was advanced by [[Joseph Bertrand|J. Bertrand]] in 1845 on the strength of tabulated data, and was proved by P.L. Chebyshev (cf. [[Chebyshev theorems on prime numbers|Chebyshev theorems on prime numbers]]). |
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Revision as of 10:24, 16 March 2023
For any natural number $n>3$ there exists a prime number that is larger than $n$ and smaller than $2n-2$. In its weaker formulation Bertrand's postulate states that for any $x>1$ there exists a prime number in the interval $(x,2x)$. The postulate was advanced by J. Bertrand in 1845 on the strength of tabulated data, and was proved by P.L. Chebyshev (cf. Chebyshev theorems on prime numbers).
References
| [1] | P.L. Chebyshev, "Oeuvres de P.L. Tchebycheff" , 1 , Chelsea, reprint (1961) (Translated from Russian) |
Comments
References
| [a1] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Clarendon Press (1965) pp. 343ff |
How to Cite This Entry:
Bertrand postulate. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bertrand_postulate&oldid=31629
Bertrand postulate. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bertrand_postulate&oldid=31629
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article