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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: gbergman@math.berkeley.edu)
Corresponding
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Abstract

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

[1]Bergman, G. M., ‘Addenda to “On common divisors of multinomial coefficients”’, unpublished note, March 2010, 9 pp., readable at http://math.berkeley.edu/∼gbergman/papers/unpub/.Google Scholar
[2]Erdős, P. and Szekeres, G., ‘Some number theoretic problems on binomial coefficients’, Aust. Math. Soc. Gaz. 5 (1978), 9799, readable at www.math-inst.hu/∼p_erdos/1978-46.pdf.Google Scholar
[3]Good, I. J., ‘Short proof of a conjecture by Dyson’, J. Math. Phys. 11 (1970), 1884 (In the second display in this telegraphic note, the final equation i=j is, as far as I can see, meaningless, and should be ignored.)CrossRefGoogle Scholar
[4]Guy, R. K., Unsolved Problems in Number Theory, 3rd edn (Springer, New York, 2004).CrossRefGoogle Scholar
[5]Granville, A., ‘Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers’, in: Organic Mathematics (Burnaby, BC, 1995), CMS Conference Proceedings, 20 (American Mathematical Society, Providence, RI, 1997), pp. 253276.Google Scholar
[6]Kummer, E. E., ‘Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen’, J. reine angew. Math. 44 (1852), 93146, readable at www.digizeitschriften.de/no_cache/en/home/, and in the author’s Collected Papers, Springer, Berlin–New York, 1975.CrossRefGoogle Scholar
[7]Schinzel, A. and Sierpiński, W., ‘Sur certaines hypothèses concernant les nombres premiers’, Acta Arith. 4 (1958), 185208; erratum at 5 (1958), 259.CrossRefGoogle Scholar
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ON COMMON DIVISORS OF MULTINOMIAL COEFFICIENTS
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