Bertrand’s postulate

  • Martin Aigner
  • Günter M. Ziegler


We have seen that the sequence of prime numbers 2, 3, 5, 7,... is infinite. To see that the size of its gaps is not bounded, let N:= 2 · 3 · 5 ·... · p denote the product of all prime numbers that are smaller than k + 2, and note that none of the k numbers
$$N + 2, N + 3, N + 4, \ldots, N + k, N + \left( {k + 1} \right)$$
is prime, since for 2 ≤ ik + 1 we know that i has a prime factor that is smaller than k + 2, and this factor also divides N, and hence also N + i. With this recipe, we find, for example, for k = 10 that none of the ten numbers
$$2312,2313,2314, \ldots,2321$$
is prime.


Prime Factor Prime Number Riemann Hypothesis Binomial Coefficient Prime Number Theorem 
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  1. [1]
    P. Erdős: Beweis eines Satzes von Tschebyschef Acta Sci. Math. (Szeged) 5 (1930-1932), 194–198.Google Scholar
  2. [2]
    R. L. Graham, D. E. Knuth & O. Patashnik: Concrete Mathematics. A Foundation for Computer Science, Addison-Wesley, Reading MA 1989.zbMATHGoogle Scholar
  3. [3]
    G. H. Hardy & E. M. Wright: An Introduction to the Theory of Numbers, fifth edition, Oxford University Press 1979.Google Scholar
  4. [4]
    P. Ribenboim: The New Book of Prime Number Records, Springer-Verlag, New York 1989.CrossRefGoogle 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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