Characterization of Chebyshev Numbers
Let Tn(x) be the degree-n Chebyshev polynomial of the first kind. It is known [1,13] that Tp(x)≡xpmodp, when p is an odd prime, and therefore, Tp(a)≡amodp for all a. Our main result is the characterization of composite numbers n satisfying the condition Tn(a)≡amodn, for any integer a. We call these...
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| Дата: | 2008 |
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| Формат: | Стаття |
| Мова: | English |
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Інститут прикладної математики і механіки НАН України
2008
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| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/152391 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Characterization of Chebyshev Numbers / D.P. Jacobs, V. Trevisan, M.O. Rayers // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 2. — С. 65–82. — Бібліогр.: 17 назв. — англ. |
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irk-123456789-1523912019-06-19T20:06:43Z Characterization of Chebyshev Numbers Jacobs, D.P. Trevisan, V. Rayers, M.O. Let Tn(x) be the degree-n Chebyshev polynomial of the first kind. It is known [1,13] that Tp(x)≡xpmodp, when p is an odd prime, and therefore, Tp(a)≡amodp for all a. Our main result is the characterization of composite numbers n satisfying the condition Tn(a)≡amodn, for any integer a. We call these pseudoprimes Chebyshev numbers, and show that n is a Chebyshev number if and only if n is odd, squarefree, and for each of its prime divisors p, n≡±1modp−1 and n≡±1modp+1. Like Carmichael numbers, they must be the product of at least three primes. Our computations show there is one Chebyshev number less than 10¹⁰, although it is reasonable to expect there are infinitely many. Our proofs are based on factorization and resultant properties of Chebyshev polynomials. 2008 Article Characterization of Chebyshev Numbers / D.P. Jacobs, V. Trevisan, M.O. Rayers // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 2. — С. 65–82. — Бібліогр.: 17 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:11A07, 11Y35. http://dspace.nbuv.gov.ua/handle/123456789/152391 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
Let Tn(x) be the degree-n Chebyshev polynomial of the first kind. It is known [1,13] that Tp(x)≡xpmodp, when p is an odd prime, and therefore, Tp(a)≡amodp for all a. Our main result is the characterization of composite numbers n satisfying the condition Tn(a)≡amodn, for any integer a. We call these pseudoprimes Chebyshev numbers, and show that n is a Chebyshev number if and only if n is odd, squarefree, and for each of its prime divisors p, n≡±1modp−1 and n≡±1modp+1. Like Carmichael numbers, they must be the product of at least three primes. Our computations show there is one Chebyshev number less than 10¹⁰, although it is reasonable to expect there are infinitely many. Our proofs are based on factorization and resultant properties of Chebyshev polynomials. |
| format |
Article |
| author |
Jacobs, D.P. Trevisan, V. Rayers, M.O. |
| spellingShingle |
Jacobs, D.P. Trevisan, V. Rayers, M.O. Characterization of Chebyshev Numbers Algebra and Discrete Mathematics |
| author_facet |
Jacobs, D.P. Trevisan, V. Rayers, M.O. |
| author_sort |
Jacobs, D.P. |
| title |
Characterization of Chebyshev Numbers |
| title_short |
Characterization of Chebyshev Numbers |
| title_full |
Characterization of Chebyshev Numbers |
| title_fullStr |
Characterization of Chebyshev Numbers |
| title_full_unstemmed |
Characterization of Chebyshev Numbers |
| title_sort |
characterization of chebyshev numbers |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/152391 |
| citation_txt |
Characterization of Chebyshev Numbers / D.P. Jacobs, V. Trevisan, M.O. Rayers // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 2. — С. 65–82. — Бібліогр.: 17 назв. — англ. |
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Algebra and Discrete Mathematics |
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