A divisibility property of binomial coefficients viewed as an elementary sieve

[摘要] The triangular array of binomial coefficients012301111212131331…is said to have undergone aj-shift if ther-th row of the triangle is shiftedrjunits to the right(r=0,1,2,…). Mann and Shanks have proved that in a 2-shifted array a column numberc>1is 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>2are considered in this paper. Moreover, an analog of the criterion of Mann and Shanks [2] is given which is valid for arbitrary arithmetic progressions.

[学科分类] 数学（综合）