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Estimates of the least prime factor of a binomial coefficient

Part of: Sequences and sets

Published online by Cambridge University Press:  26 February 2010

S. V. Konyagin
S. V. Konyagin
Affiliation:
Dept. of Mech. and Math., Moscow State University, Moscow, 119899, Russia. e-mail: kon@nw.math.msu.su
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Abstract

Let k be a positive integer and g(k) be the least integer n > k + 1 such that all prime factors of are >k. We prove that

where c is an absolute positive constant. We also establish a new theorem on the distribution of fractional parts of a smooth function.

MSC classification

Secondary: 11B65: Binomial coefficients; factorials; $q$-identities
Type
Research Article
Information
Mathematika , Volume 46 , Issue 1 , June 1999 , pp. 41 - 55
Copyright
Copyright © University College London 1999

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References

1.Baker, R. C. and Harman, G.The difference between consecutive primes Proc. Lond. Math. Soc. (3) 72 (1986), 261–280.Google Scholar
2.Bombieri, E. and Pila, J.The number of integral points on arcs and ovals Duke Math. J., 59 (1989), 337–357.CrossRefGoogle Scholar
3.Ecklund, E. F., Erdős, P. and Selfridge, J. L.A new function associated with the prime factors of Math. Comp, 28(1974), 647–649.Google Scholar
4.Erdős, P., Lacampagne, C. B. and Selfridge, J. L.Estimates of the least prime factor of a binomial coefficient. Math. Comp., 61(1993), 215–224.CrossRefGoogle Scholar
5.Filaseta, M. and Trifonov, O.The distribution of fractional parts with applications to gap results in number theory. Proc. Lond. Math. Soc., 73(1996), 241–278.CrossRefGoogle Scholar
6.Granville, A. and Ramaré, O.. Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. Mathematika, 43(1996), 73–107.CrossRefGoogle Scholar
7.Huxley, M. N. and Trifonov, O.. The square-full numbers in an interval. Math. Proc. Cambr. Phil. Soc., 119(1996), 201–208.CrossRefGoogle Scholar
8.Sander, J.. On a sum over primes. Hardy-Ramanujan J., 17(1994), 32–39.Google Scholar
9.Schmidt, W. M.. Diophantine approximation and diophantine equations. Led. Notes in Math.. vol. 1467 (Springer-Verlag, 1991).CrossRefGoogle Scholar
10.Swinnerton-Dyer, H. P. F.. The number of lattice points on a convex curve. J. Number Theory, 6(1974), 128–135.CrossRefGoogle Scholar