International Journal of Mathematics and Mathematical Sciences
Volume 4 (1981), Issue 4, Pages 731-743
doi:10.1155/S0161171281000562

A divisibility property of binomial coefficients viewed as an elementary sieve

Richard H. Hudson1 and Kenneth S. Williams2

1Department of Mathematics, Computer Science, and Statistics University of South Carolina, Columbia 29208, South Carolina, USA
2Department of Mathematics, Carleton University, Ottawa KIS 5B6, Ontario, Canada

Received 2 December 1981

Copyright © 1981 Richard H. Hudson and Kenneth S. Williams. 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.

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.