Research Article

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

Andrew Granvillea1 and Olivier Ramaréa2

a1 Department of Mathematics, University of Georgia, Athens, Georgia 30602, U.S.A. e-mail andrew@math.uga.edu

a2 Université Nancy I, 54506 Vandoeuvre-les-Nancy, France. e-mail ramare@iecn.u-nancy.fr

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 S0025579300011608_inline1 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 S0025579300011608_inline1 is squarefree. One well-known conjecture, due to Erdős, is that S0025579300011608_inline2 is not squarefree once n > 4. Sarközy [Sz] proved this for sufficiently large n but here we return to and solve the original question.

(Received June 20 1994)

Key Words:

  • 11B65: NUMBER THEORY; Sequences and Sets; Binomial coefficients.