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Bertrand’s postulate

  • Martin Aigner
  • Günter M. Ziegler

Abstract

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.

Keywords

Prime Factor Prime Number Riemann Hypothesis Binomial Coefficient Prime Number Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  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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