Ukrainian Mathematical Journal

, Volume 59, Issue 12, pp 1914–1921 | Cite as

Re-extending Chebyshev’s theorem about Bertrand’s conjecture

  • A. Shams
Brief Communications


In this paper, Chebyshev’s theorem (1850) about Bertrand’s conjecture is re-extended using a theorem about Sierpinski’s conjecture (1958). The theorem had been extended before several times, but this extension is a major extension far beyond the previous ones. At the beginning of the proof, maximal gaps table is used to verify initial states. The extended theorem contains a constant r, which can be reduced if more initial states can be checked. Therefore, the theorem can be even more extended when maximal gaps table is extended. The main extension idea is not based on r, though.


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  1. 1.
    G. Mosaheb, Elementary Number Theory, Vol. 2, Pt. 2, Soroush Press, Tehran (1980).Google Scholar
  2. 2.
    P. Ribenboim, The New Book of Prime Number Records, Springer, New York (1995).Google Scholar
  3. 3.
  4. 4.
    J. Young and A. Potler, “First occurrence of prime gaps,” Math. Comput., 53, 221–224. (1989).MathSciNetGoogle Scholar
  5. 5.
    T. Nicely, “New maximal prime gaps and first occurrence,” Math. Comput., 68, 1311–1315 (1999).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    T. Nicely and B. Nyman, First Occurrence of a Prime Gap of 1000 or Greater, Preprint,

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • A. Shams
    • 1
    • 2
  1. 1.The University of ManchesterManchesterUK
  2. 2.Ferdowsi University of MashhadMashhadIran

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