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ON COMMON DIVISORS OF MULTINOMIAL COEFFICIENTS

  • GEORGE M. BERGMAN (a1)

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.

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References

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[1]Bergman, G. M., ‘Addenda to “On common divisors of multinomial coefficients”’, unpublished note, March 2010, 9 pp., readable at http://math.berkeley.edu/∼gbergman/papers/unpub/.
[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.
[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.)
[4]Guy, R. K., Unsolved Problems in Number Theory, 3rd edn (Springer, New York, 2004).
[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.
[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.
[7]Schinzel, A. and Sierpiński, W., ‘Sur certaines hypothèses concernant les nombres premiers’, Acta Arith. 4 (1958), 185208; erratum at 5 (1958), 259.
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ON COMMON DIVISORS OF MULTINOMIAL COEFFICIENTS

  • GEORGE M. BERGMAN (a1)

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