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ON COMMON DIVISORS OF MULTINOMIAL COEFFICIENTS

Published online by Cambridge University Press:  13 October 2010

GEORGE M. BERGMAN*
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA (email: [email protected])
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Abstract

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Erdős and Szekeres [‘Some number theoretic problems on binomial coefficients’, Aust. Math. Soc. Gaz.5 (1978), 97–99] showed that for any four positive integers satisfying m1+m2=n1+n2, the two binomial coefficients (m1+m2)!/m1!m2! and (n1+n2)!/n1!n2! have a common divisor greater than 1. The analogous statement for k-element families of k-nomial coefficients (k>1) was conjectured in 1997 by David Wasserman.

Erdős and Szekeres remark that if m1,m2,n1,n2 as above are all greater than 1, there is probably a lower bound on the common divisor in question which goes to infinity as a function of m1 +m2 . Such a bound is obtained in Section 2.

The remainder of this paper is devoted to proving results that narrow the class of possible counterexamples to Wasserman’s conjecture.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

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