Difference between revisions of "Bertrand postulate"
From Encyclopedia of Mathematics
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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]]). | ||
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.L. Chebyshev, | + | <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> | |
| − | + | <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> | |
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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=12676
Bertrand postulate. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bertrand_postulate&oldid=12676
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article