A divisibility property of binomial coefficients viewed as an elementary sieve
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 t...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1981-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171281000562 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |