For any k â‰¥ 1, let pk be the kth prime number. In this paper, we confirm a conjecture of ErdÅ‘s and Stewart concerning all the solutions of the diophantine equation n! + 1 = pkp b k+1, when pkâˆ’1 â‰¤ n <â€¦ (More)

For every positive integer n let p(n) be the largest prime number p â‰¤ n. Given a positive integer n = n1, we study the positive integer r = R(n) such that if we define recursively ni+1 = ni âˆ’ p(ni)â€¦ (More)

Radiology is among the specialties with an increasing risk of litigation. Regardless of the outcome of legal proceedings, physicians who are sued usually perceive the claim as an assault on theirâ€¦ (More)

For a positive integer n and its Euler function Ï†(n) we write Ï†(n)/n = a/b, where a = a(n) and b = b(n) are coprime. For a fixed integer a, we consider the number of integers b for which the aboveâ€¦ (More)

Miyaji, Nakabayashi and Takano have recently suggested a construction of the so-called MNT elliptic curves with low embedding degree, which are also of importance for pairing-based cryptography. Weâ€¦ (More)

We discuss the smallest algebraic number field which contains the nth roots of unity and which may be reached from the rational field Q by a sequence of irreducible, radical, Galois extensions. Theâ€¦ (More)