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
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.
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