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Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients

Part of: Sequences and sets

Published online by Cambridge University Press:  26 February 2010

Andrew Granville  and
Olivier Ramaré
Andrew Granville
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602, U.S.A. e-mail andrew@math.uga.edu
Olivier Ramaré
Affiliation:
Université Nancy I, 54506 Vandoeuvre-les-Nancy, France. e-mail ramare@iecn.u-nancy.fr
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Extract

The distribution of squarefree binomial coefficients. For many years, Paul Erdős has asked intriguing questions concerning the prime divisors of binomial coefficients, and the powers to which they appear. It is evident that, if k is not too small, then must be highly composite in that it contains many prime factors and often to high powers. It is therefore of interest to enquire as to how infrequently is squarefree. One well-known conjecture, due to Erdős, is that is not squarefree once n > 4. Sarközy [Sz] proved this for sufficiently large n but here we return to and solve the original question.

MSC classification

Secondary: 11B65: Binomial coefficients; factorials; $q$-identities
Type
Research Article
Information
Mathematika , Volume 43 , Issue 1 , June 1996 , pp. 73 - 107
Copyright
Copyright © University College London 1996

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