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Korselt number — 1 results found.

      
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On the Korselt Set of a Squarefree Composite Number
C. R. Math. Rep. Acad. Sci. Canada Vol. 35 (1) 2013, pp. 1–15
Ibrahim Al-Rasasi; Othman Echi; Nejib Ghanmi (Received: 2012/09/01, Revised: 2012/11/18)

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Let \(\alpha\in \mathbb{Z}\setminus \{0\}\). A positive composite squarefree integer \(N\) is said to be an \(\alpha\)-Korselt number (\(K_{\alpha}\)-number, for short) if \(N\neq \alpha\) and \(p-\alpha\) divides \(N-\alpha\) for each prime divisor \(p\) of \(N\). By the Korselt set of \(N\), we mean the set of all \(\alpha\in \mathbb{Z}\setminus \{0\}\) such that \(N\) is a \(K_{\alpha}\)-number. This set will be denoted by \(\mathcal{KS}(N)\).

In a recent paper , Bouallegue–Echi–Pinch have asked whether there are infinitely many squarefree composite numbers with empty Korselt set. This paper aims to solve this question by showing that for each prime number \(q\geq 19\), \(6q\) has an empty Korselt set.

We also show that for each integer \(l\geq 3\), there are infinitely many squarefree composite numbers with \(l\) prime divisors whose Korselt sets are empty.

Soit \(N\) un nombre composé sans facteur carré et \(\alpha\in \mathbb{Z} \setminus \{0\}\). On dit que \(N\) est \(\alpha\)-Korselt si \(N\neq \alpha\) et \(p-\alpha\) divise \(N-\alpha\) pour tout facteur premier \(p\) de \(N\).

L’ensemble constitué de tous les \(\alpha\) tels que \(N\) est \(\alpha\)-Korselt, noté \(\mathcal{KS}(N)\), est appelé l’ensemble de Korselt de \(N\).

Bouallegue–Echi–Pinch se sont posés la question d’existence d’une infinité de nombres composés sans facteur carré possédant des ensembles de Korselt vides.

Dans ce papier on donne une réponse positive à cette question en démontrant que pour tout premier \(q\geq 19\), \(6q\) a un ensemble de Korselt vide.

On prouve aussi que pour tout entier \(l\geq 3\), il existe une infinité de nombres composés sans facteur carré ayant \(l\) facteurs premiers et d’ensembles de Korselt vides.

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algebraic number theory approximation property automorphisms Bessel functions Boson-fermion correspondence C*-algebra Carmichael number center problem Chebyshev transform classification Classification of simple C*-algebras composition operators continued fractions Cuntz Semigroup elliptic curves fixed point Fourier transform function fields. functoriality general relativity generic property ideals indefinite inner product inductive limits of sub-homogeneous C*- algebras Irrational rotation algebra J-Hermitian matrix K-theory Kahler manifolds L-functions maximal ideal space nonexpansive mapping numerical range orthogonal polynomials Predual space prime number property SP Renormalization rotation algebras Salem number semi-reciprocal polynomials tracially approximate splitting interval algebras unbounded traces uniqueness Weak Markov set Whitney problems

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