International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1981 / Article

Open Access

Volume 4 |Article ID 856269 | https://doi.org/10.1155/S0161171281000562

Richard H. Hudson, Kenneth S. Williams, "A divisibility property of binomial coefficients viewed as an elementary sieve", International Journal of Mathematics and Mathematical Sciences, vol. 4, Article ID 856269, 13 pages, 1981. https://doi.org/10.1155/S0161171281000562

A divisibility property of binomial coefficients viewed as an elementary sieve

Received02 Dec 1981

Abstract

The triangular array of binomial coefficients 012301111212131331 is said to have undergone a j-shift if the r-th row of the triangle is shifted rj units to the right (r=0,1,2,). Mann and Shanks have proved that in a 2-shifted array a column number c>1 is prime if and only if every entry in the c-th column is divisible by its row number. Extensions of this result to j-shifted arrays where j>2 are considered in this paper. Moreover, an analog of the criterion of Mann and Shanks [2] is given which is valid for arbitrary arithmetic progressions.

Copyright © 1981 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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