Q -Korselt numbers

Abstract

Let α = α1 α2 ∈ Q \ {0} ; a positive integer N is said to be an α-Korselt number (Kα-number, for short) if N ̸ = α and α2p − α1 divides α2N − α1 for every prime divisor p of N. In this paper we prove that for each squarefree composite number N there exist finitely many rational numbers α such that N is a Kα-number and if α ≤ 1 then N has at least three prime factors. Moreover, we prove that for each α ∈ Q \ {0} there exist only finitely many squarefree composite numbers N with two prime factors such that N is a Kα-number.

Numbers with empty rational Korselt sets

“…Since α-Korselt numbers for α ∈ Z (or simply Korselt numbers) were introduced, they have been the subject of intensive study, one may find more details in [1,2,6,7]. Motivated by these facts, Ghanmi [4] introduced the notion of Q-Korselt numbers as extension of Korselt numbers to Q by setting the following definitions.…”

“…It's clear that every nonzero positive integer has finitely many Korselt bases over Q (see [4,Theorem 2.3]), hence a natural question can be posed about the existence of such numbers with empty rational Korselt set and how many there are. Obviously, this cannot happen for N = pq; numbers with two distinct primes factors because q + p − 1 lies always in Q-KS(N ).…”

Connections on the rational Korselt set of $pq$

“…The α-Korselt numbers for α ∈ Z have been thoroughly investigated in recent years, especially in [1,3,4,8,9]. In [5], Ghanmi proposed another generalization for α = α 1 α 2 ∈ Q\{0} by setting the following definitions:…”

“…The simple case when N = pq is still full of unsolved problems. For instance, after examining the Korselt sets over Q of some values of N = pq, since Q-KW(N ) is finite (see [5,Theorem 2.3]), we state the following conjecture: Conjecture 4.7. For all N = pq, Q-KW(N ) is odd.…”

Connections on the Rational Korselt Set of pq

“…The α-Korselt numbers for α ∈ Z are well investigated last years specially in [2,3,6,10,11]. In [1], Ghanmi proposed another generalization for α = α 1 α 2 ∈ Q \ {0} by setting the following definitions.…”