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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| Published in: | Algebra and Discrete Mathematics |
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| Date: | 2008 |
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| Format: | Article |
| Language: | English |
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Інститут прикладної математики і механіки НАН України
2008
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/152391 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | 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| _version_ | 1862743271956021248 |
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| author | Jacobs, D.P. Trevisan, V. Rayers, M.O. |
| author_facet | Jacobs, D.P. Trevisan, V. Rayers, M.O. |
| 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. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| 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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| first_indexed | 2025-12-07T20:29:06Z |
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| id | nasplib_isofts_kiev_ua-123456789-152391 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T20:29:06Z |
| publishDate | 2008 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Jacobs, D.P. Trevisan, V. Rayers, M.O. 2019-06-10T19:03:32Z 2019-06-10T19:03:32Z 2008 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. https://nasplib.isofts.kiev.ua/handle/123456789/152391 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. Research partially supported by CNPq - Grants 478290/04-7 and 43991/2005-0;and FAPERGS - Grant 05/2024.1 en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Characterization of Chebyshev Numbers Article published earlier |
| spellingShingle | Characterization of Chebyshev Numbers Jacobs, D.P. Trevisan, V. Rayers, M.O. |
| title | Characterization of Chebyshev Numbers |
| title_full | Characterization of Chebyshev Numbers |
| title_fullStr | Characterization of Chebyshev Numbers |
| title_full_unstemmed | Characterization of Chebyshev Numbers |
| title_short | Characterization of Chebyshev Numbers |
| title_sort | characterization of chebyshev numbers |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/152391 |
| work_keys_str_mv | AT jacobsdp characterizationofchebyshevnumbers AT trevisanv characterizationofchebyshevnumbers AT rayersmo characterizationofchebyshevnumbers |