The $${\mathbb {Q}}$$ Q -Korselt set of $$\mathrm {pq}$$ pq
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The Q-Korselt set of pq Nejib Ghanmi1,2
© Akadémiai Kiadó, Budapest, Hungary 2020
Abstract α1 Let N be a positive integer, A be a nonempty subset of Q and α = ∈ A\{0, N }. α is α2 called an N -Korselt base (equivalently N is said an α-Korselt number) if α2 p − α1 is a divisor of α2 N − α1 for every prime p dividing N . The set of all Korselt bases of N in A is called the A-Korselt set of N and is simply denoted by A-KS (N ). Let p and q be two distinct prime numbers. In this paper, we study the Q-Korselt bases of pq, where we give in detail how to provide Q-KS ( pq). Consequently, we finish the incomplete characterization of the Korselt set of pq over Z given in Ghanmi (JAMA 42:2752–2762, 2018), by supplying the set Z-KS ( pq) when q < 2 p. Keywords Prime number · Carmichael number · Square free composite number · Korselt base · Korselt number · Korselt set Mathematics Subject Classification Primary 11Y 16; Secondary 11Y 11 · 11A51
1 Introduction The notion of Korselt numbers ( or α-Korselt number with α ∈ Z) was introduced by Bouallègue–Echi–Pinch [2,7] as a generalization of Carmichael numbers [1,3]. Korselt numbers are defined simply as numbers which meet a generalized Korselt criterion as follows. Definition 1.1 [8] Let α ∈ Z\{0}. A positive integer N is said to be an α-Korselt number (K α -number, for short) if N = α and p − α divides N − α for each prime divisor p of N . Recently, considerable progress has been made investigating Korselt numbers especially in [2,4,5,7]. Many properties of Carmichael numbers are extended for Korselt numbers. However, many related questions remain open until now, such as the infinitude of Korselt numbers, providing a complete characterization of the Korselt set of such a number, etc.
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Nejib Ghanmi [email protected]; [email protected]
1
Preparatory Institute of Engineering Studies, Tunis University, Tunis, Tunisia
2
Department of Mathematics, University College of Jammum, Mekkah, Saudi Arabia
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N. Ghanmi
Recently, Ghanmi proposed in [5,6] another generalization of Carmichael numbers; he extended the notion of Korselt numbers to Q by stating the following definitions. Definition 1.2 [5] Let N ∈ N\{0, 1}, α =
α1 ∈ Q\{0} and A be a subset of Q. Then α2
(1) N is said to be an α-Korselt number (K α -number, for short), if N = α and α2 p − α1 divides α2 N − α1 for every prime divisor p of N . (2) By the A-Korselt set of the number N (or the Korselt set of N over A) , we mean the set A-KS (N ) of all β ∈ A\{0, N } such that N is a K β -number. (3) The cardinality of A-KS (N ) will be called the A-Korselt weight of N ; we denote it by A-KW (N ). It is obvious by this definition, that for α ∈ Z\{0} (i.e. α2 = 1) we obtain the original α-Korselt numbers introduced by Bouallègue–Echi–Pinch [2]. Definition 1.3 [6] Let N ∈ N\{0, 1}, α ∈ Q and B be a subset of N. Then (1) α is called N -Korselt base (K N -base, for short), if N is a K α -number. (2) By the B-Korselt set of the base α (or the Korselt set of the base α over B), we mean the set B-KS (B(α)) of all M ∈ B su
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