Characterization of Chebyshev Numbers
Vernadsky National Library of Ukraine
Переглянути архів Інформація| Поле | Співвідношення | |
| Title |
Characterization of Chebyshev Numbers
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| Creator |
Jacobs, D.P.
Trevisan, V. Rayers, M.O. |
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| 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.
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| Date |
2019-06-10T19:03:32Z
2019-06-10T19:03:32Z 2008 |
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| Type |
Article
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| Identifier |
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 |
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| Language |
en
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| Relation |
Algebra and Discrete Mathematics
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| Publisher |
Інститут прикладної математики і механіки НАН України
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