Intervals containing prime numbers

Laurențiu Panaitopol
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 8, 2002, Number 4, Pages 145—148
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Authors and affiliations

Laurențiu Panaitopol
University of Bucharest, Faculty of Mathematics
14 Academiei St., RO-70109 Bucharest, Romania

Abstract

For x > 0, let π(x) be the number of prime numbers not exceeding x. One shows that, for x > 7, there exists at least one prime number between x and x + π(x), thus obtaining a result that is sharper than the one postulated by Bertrand.

Keywords

  • distribution of prime numbers, inequalities.Bertrand’s postulate

AMS Classification

  • 11A35
  • 11N05

References

  1. Bertrand .J., Memoire sur le nombre de valeurs que pent prendre une function quand on y permute les lettres qu’elle renferme. J. L’Ecole Royale Polytechn. 18 (1845), 123-140.
  2. Chebyshev P.L., Memoire sur le nombres premiers. J. Math. Pures Appl. 17 (1852), 366-390.
  3. Costa Pereira N., Elementary estimate for the Chebyshev function 4/(x) and the Mobius function M(x). Acta Arith. 52 (1989), 307-337.
  4. Dusart P., Inegalites explicites pour ‘I'(x), Q(x), n(x) et les nombres premiers. C. R. Math. Acad. Sci. Soc. R. Can. 2 (1999), 53-59.
  5. Mitrinovic D.S., Sandor J., Crstici B., Handbook of Number Theory. Kluwer Academic Publishers, Dordrecht-Boston-London, 1996.
  6. Nagura J., On the interval containing at least one prime number. Proc. Japan. Acad. 28 (1952), 177-181.
  7. Panaitopol L., A special case of the hardy-Littlewood conjecture. Math. Reports (to appear).
  8. Rohrbach H., Weis J., Zum finiten Fall des Bertrandschen Postulats. J. reine angew. Math. 214/215 (1964), 432-440.
  9. Rosser J.B., Schoenfeld L., Approximate formulas for functions of prime numbers. Illinois J. Math. 6 (1962), 64-94.

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Cite this paper

APA

Panaitopol, L. (2002). Intervals containing prime numbers. Notes on Number Theory and Discrete Mathematics, 8(4), 145-148.

Chicago

Panaitopol, L. “Intervals containing prime numbers.” Notes on Number Theory and Discrete Mathematics 8, no. 4 (2002): 145-148.

MLA

Panaitopol, L. “Intervals containing prime numbers.” Notes on Number Theory and Discrete Mathematics 8.4 (2002): 145-148. Print.

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