International Journal of Mathematics and Mathematical Sciences
Volume 4 (1981), Issue 4, Pages 731-743
A divisibility property of binomial coefficients viewed as an elementary sieve
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.
The triangular array of binomial coefficients
is said to have undergone a -shift if the -th row of the triangle is shifted units to the right . Mann and Shanks have proved that in a 2-shifted array a column number is prime if and only if every entry in the -th column is divisible by its row number. Extensions of this result to -shifted arrays where are considered in this paper. Moreover, an analog of the criterion of Mann and Shanks  is given which is valid for arbitrary arithmetic progressions.