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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Lugansk National Taras Shevchenko University
2018
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua: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 |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
Chebyshev polynomials polynomial factorization resultant pseudoprimes Carmichael numbers 11A07 11Y35 |
| spellingShingle |
Chebyshev polynomials polynomial factorization resultant pseudoprimes Carmichael numbers 11A07 11Y35 Jacobs, David Pokrass Rayes, Mohamed O. Trevisan, Vilmar Characterization of Chebyshev Numbers |
| topic_facet |
Chebyshev polynomials polynomial factorization resultant pseudoprimes Carmichael numbers 11A07 11Y35 |
| format |
Article |
| author |
Jacobs, David Pokrass Rayes, Mohamed O. Trevisan, Vilmar |
| author_facet |
Jacobs, David Pokrass Rayes, Mohamed O. Trevisan, Vilmar |
| author_sort |
Jacobs, David Pokrass |
| 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 |
| 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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/809 |
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2024-04-12T06:25:25Z |
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