Intervals containing prime numbers

L. Panaitopol
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 7, 2001, Number 4, Pages 111–114
Full paper (PDF, 122 Kb)

Details

Authors and affiliations

L. 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 peut 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 ‘I'(x) and the Mobius function M(x). Acta Arith. 52 (1989), 307-337.
4. Dusart P., Inegafites explicites pour ‘F(x), Q(x), 7r(x) et les nombres premiers. C. R. Math. Acad. Sci. Soc. R. Can. 2 (1999), 53-59.
5. Mitrinovic D.S., Sándor 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.