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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| Veröffentlicht in: | Algebra and Discrete Mathematics |
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| Datum: | 2008 |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут прикладної математики і механіки НАН України
2008
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/152391 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | 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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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Zusammenfassung: | 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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| ISSN: | 1726-3255 |