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  • 1 Preparatory Institut of Engineering Studies of Tunis, Montfleury, Tunisia
  • 2 University College of Jammum, Department of Mathematics, Saudi Arabia
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Abstract

Let N be a positive integer, A be a subset of ℚ and α=α1α2A\{0,N}. N is called an α-Korselt number (equivalently α is said an N-Korselt base) if α 2 pα 1 divides α 2 Nα 1 for every prime divisor p of N. By the Korselt set of N over A, we mean the set AKS(N) of all αA\{0,N} such that N is an α-Korselt number.

In this paper we determine explicitly for a given prime number q and an integer l ∈ ℕ \{0, 1}, the set A-KS(ql) and we establish some connections between the ql -Korselt bases in ℚ and others in ℤ. The case of A=[1,1[ is studied where we prove that ([1,1[)-KS(ql) is empty if and only if l = 2.

Moreover, we show that each nonzero rational α is an N-Korselt base for infinitely many numbers N = ql where q is a prime number and l ∈ ℕ.

  • [1]

    Beeger, N. G. W. H., On composite numbers n for which an −1 ≡ 1 (mod n) for every a prime to n, Scripta Math., 16 (1950), 133135.

    • Search Google Scholar
    • Export Citation
  • [2]

    Bouallegue, K., Echi, O. and Pinch, R., Korselt Numbers and Sets, Int. J. Number Theory, 6 (2010), 257269.

  • [3]

    Carmichael, R. D., Note on a new number theory function, Bull. Amer. Math. Soc., 16 (1910), 232238.

  • [4]

    Carmichael, R. D., On composite numbers P which satisfy the Fermat congruence aP −1 ≡ 1 (mod P), Amer. Math. Monthly, 19 (1912), 2227.

    • Search Google Scholar
    • Export Citation
  • [5]

    Echi, O., Williams Numbers, Mathematical Reports of the Academy of Sciences of the Royal Society of Canada, 29 (2) (2007), 4147.

  • [6]

    Echi, O. and Ghanmi, N., The Korselt Set of pq, Int. J. Number Theory., 8 (2) (2012), 299309.

  • [7]

    El-Rassasi, I., The Korselt set of the square of a prime, Int. J. Number Theory, 10 (4) (2014), 875884.

  • [8]

    Ghanmi, N., ℚ-Korselt Numbers, Turk. J. Math, 42 (2018), 27522762.

  • [9]

    Korselt, A., Problème chinois, L'intermediaire des Mathématiciens, 6 (1899), 142143.