Characterization of Chebyshev Numbers
Let \(T_n(x)\) be the degree-\(n\) Chebyshev polynomial of the first kind. It is known [1,13] that \(T_p(x) \equiv x^p \bmod{p}\), when \(p\) is an odd prime, and therefore, \(T_p(a) \equiv a \bmod{p}\) for all \(a\). Our main result is the characterization of composite numbers \(n\) satisfying the...
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| Дата: | 2018 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/809 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| _version_ | 1856543041694728192 |
|---|---|
| author | Jacobs, David Pokrass Rayes, Mohamed O. Trevisan, Vilmar |
| author_facet | Jacobs, David Pokrass Rayes, Mohamed O. Trevisan, Vilmar |
| author_sort | Jacobs, David Pokrass |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-03-22T09:39:19Z |
| description | Let \(T_n(x)\) be the degree-\(n\) Chebyshev polynomial of the first kind. It is known [1,13] that \(T_p(x) \equiv x^p \bmod{p}\), when \(p\) is an odd prime, and therefore, \(T_p(a) \equiv a \bmod{p}\) for all \(a\). Our main result is the characterization of composite numbers \(n\) satisfying the condition \(T_n(a) \equiv a \bmod{n}\), 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 \(p\), \(n \equiv \pm 1 \bmod p-1\) and \(n \equiv \pm 1 \bmod p+1\). Like Carmichael numbers, they must be the product of at least three primes. Our computations show there is one Chebyshev number less than \(10^{10}\), although it is reasonable to expect there are infinitely many. Our proofs are based on factorization and resultant properties of Chebyshev polynomials. |
| first_indexed | 2025-12-02T15:41:03Z |
| format | Article |
| id | admjournalluguniveduua-article-809 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:41:03Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-8092018-03-22T09:39:19Z Characterization of Chebyshev Numbers Jacobs, David Pokrass Rayes, Mohamed O. Trevisan, Vilmar Chebyshev polynomials, polynomial factorization, resultant, pseudoprimes, Carmichael numbers 11A07, 11Y35 Let \(T_n(x)\) be the degree-\(n\) Chebyshev polynomial of the first kind. It is known [1,13] that \(T_p(x) \equiv x^p \bmod{p}\), when \(p\) is an odd prime, and therefore, \(T_p(a) \equiv a \bmod{p}\) for all \(a\). Our main result is the characterization of composite numbers \(n\) satisfying the condition \(T_n(a) \equiv a \bmod{n}\), 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 \(p\), \(n \equiv \pm 1 \bmod p-1\) and \(n \equiv \pm 1 \bmod p+1\). Like Carmichael numbers, they must be the product of at least three primes. Our computations show there is one Chebyshev number less than \(10^{10}\), although it is reasonable to expect there are infinitely many. Our proofs are based on factorization and resultant properties of Chebyshev polynomials. Lugansk National Taras Shevchenko University 2018-03-22 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/809 Algebra and Discrete Mathematics; Vol 7, No 2 (2008) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/809/339 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | Chebyshev polynomials polynomial factorization resultant pseudoprimes Carmichael numbers 11A07 11Y35 Jacobs, David Pokrass Rayes, Mohamed O. Trevisan, Vilmar Characterization of Chebyshev Numbers |
| 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 |
| topic | Chebyshev polynomials polynomial factorization resultant pseudoprimes Carmichael numbers 11A07 11Y35 |
| topic_facet | Chebyshev polynomials polynomial factorization resultant pseudoprimes Carmichael numbers 11A07 11Y35 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/809 |
| work_keys_str_mv | AT jacobsdavidpokrass characterizationofchebyshevnumbers AT rayesmohamedo characterizationofchebyshevnumbers AT trevisanvilmar characterizationofchebyshevnumbers |