Binomial coefficients are (almost) never powers

  • Martin Aigner
  • Günter M. Ziegler


There is an epilogue to Bertrand’s postulate which leads to a beautiful result on binomial coefficients. In 1892 Sylvester strengthened Bertrand’s postulate in the following way:

If n ≥ 2k, then at least one of the numbers n, n − 1,..., nk + 1 has a prime divisor p greater than k.


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  1. [1]
    P. Erdős: A theorem of Sylvester and Schur, J. London Math. Soc. 9 (1934) 282–288.CrossRefGoogle Scholar
  2. [2]
    P. Erdős: On a diophantine equation, J. London Math. Soc. 26 (1951) 176–178.MathSciNetCrossRefGoogle Scholar
  3. [3]
    J. J. Sylvester: On arithmetical series, Messenger of Math. 21 (1892), 1–19 87-120; Collected Mathematical Papers Vol. 4, 1912, 687-731.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Martin Aigner
    • 1
  • Günter M. Ziegler
    • 2
  1. 1.Institut für Mathematik II (WE2)Freie Universität BerlinBerlinGermany
  2. 2.Institut für Mathematik, MA 6-2Technische Universität BerlinBerlinGermany

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