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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]]).
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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 [[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]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,   "Oeuvres de P.L. Tchebycheff" , '''1''' , Chelsea, reprint  (1961)  (Translated from Russian)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev, "Oeuvres de P.L. Tchebycheff" , '''1''' , Chelsea, reprint  (1961)  (Translated from Russian)</TD></TR>
 
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Clarendon Press  (1965)  pp. 343ff</TD></TR>
 
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</table>
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Clarendon Press  (1965)  pp. 343ff</TD></TR></table>
 

Latest revision as of 15:15, 10 April 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)
[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
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article