A divisibility property of binomial coefficients viewed as an elementary sieve. (1981)

• Record Type: Journal Article
• Authors: Hudson, Richard H.; Williams, Kenneth S.
• Is Part Of: International journal of mathematics and mathematical sciences. Volume 4:Number 4(1981)
• Page Start: 731
• Page End: 743
• Publication Date: 1981
• Subjects: array of binomial coefficients -- primes in arithmetic progressions; Mathematics -- Periodicals; 510.5
• DOI: 10.1155/S0161171281000562
• Languages: English
• ISSNs: 0161-1712

Abstract

Abstract : The triangular array of binomial coefficients 0 1 2 3 0 1 1 1 1 2 1 2 1 3 1 3 3 1 … is said to have undergone aj -shift if ther -th row of the triangle is shiftedr j units to the right( r = 0, 1, 2, … ) . Mann and Shanks have proved that in a 2-shifted array a column numberc > 1 is prime if and only if every entry in thec -th column is divisible by its row number. Extensions of this result toj -shifted arrays wherej > 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.