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On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

Given an integer n 3 , let u 1 , ... , u n be pairwise coprime integers 2 , 𝒟 a family of nonempty proper subsets of { 1 , ... , n } with “enough” elements, and ε a function 𝒟 { ± 1 } . Does there exist at least one prime q such that q divides i I u i - ε ( I ) for some I 𝒟 , but it does not divide u 1 u n ? We answer this question in the positive when the u i are prime powers and ε and 𝒟 are subjected to certain restrictions.We use the result to prove that, if ε 0 { ± 1 } and A is a set of three or more primes that contains all prime divisors of any number of the form p B p - ε 0 for...

On certain arithmetic functions involving the greatest common divisor

Ekkehard Krätzel, Werner Nowak, László Tóth (2012)

Open Mathematics

The paper deals with asymptotics for a class of arithmetic functions which describe the value distribution of the greatest-common-divisor function. Typically, they are generated by a Dirichlet series whose analytic behavior is determined by the factor ζ2(s)ζ(2s − 1). Furthermore, multivariate generalizations are considered.

On the least common multiple of Lucas subsequences

Shigeki Akiyama, Florian Luca (2013)

Acta Arithmetica

We compare the growth of the least common multiple of the numbers u a 1 , . . . , u a n and | u a 1 u a n | , where ( u n ) n 0 is a Lucas sequence and ( a n ) n 0 is some sequence of positive integers.

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