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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| Format: | Article |
| Language: | English |
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Wiley
1981-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171281000562 |
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| author | Richard H. Hudson Kenneth S. Williams |
| author_facet | Richard H. Hudson Kenneth S. Williams |
| author_sort | Richard H. Hudson |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-f253c92cae0f4f3986e174eaf32bcb2b |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1981-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-f253c92cae0f4f3986e174eaf32bcb2b2025-02-03T01:02:56ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251981-01-014473174310.1155/S0161171281000562A divisibility property of binomial coefficients viewed as an elementary sieveRichard H. Hudson0Kenneth S. Williams1Department of Mathematics, Computer Science, and Statistics University of South Carolina, Columbia 29208, South Carolina, USADepartment of Mathematics, Carleton University, Ottawa KIS 5B6, Ontario, CanadaThe 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.http://dx.doi.org/10.1155/S0161171281000562array of binomial coefficientsprimes in arithmetic progressions. |
| spellingShingle | Richard H. Hudson Kenneth S. Williams A divisibility property of binomial coefficients viewed as an elementary sieve International Journal of Mathematics and Mathematical Sciences array of binomial coefficients primes in arithmetic progressions. |
| title | A divisibility property of binomial coefficients viewed as an elementary sieve |
| title_full | A divisibility property of binomial coefficients viewed as an elementary sieve |
| title_fullStr | A divisibility property of binomial coefficients viewed as an elementary sieve |
| title_full_unstemmed | A divisibility property of binomial coefficients viewed as an elementary sieve |
| title_short | A divisibility property of binomial coefficients viewed as an elementary sieve |
| title_sort | divisibility property of binomial coefficients viewed as an elementary sieve |
| topic | array of binomial coefficients primes in arithmetic progressions. |
| url | http://dx.doi.org/10.1155/S0161171281000562 |
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