33 research outputs found
Coincidences in generalized Lucas sequences
For an integer , let be the generalized
Lucas sequence which starts with ( terms) and each term
afterwards is the sum of the preceding terms. In this paper, we find all
the integers that appear in different generalized Lucas sequences; i.e., we
study the Diophantine equation in nonnegative integers
with . The proof of our main theorem uses lower
bounds for linear forms in logarithms of algebraic numbers and a version of the
Baker-Davenport reduction method. This paper is a continuation of the earlier
work [4].Comment: 14 page
Some arithmetic functions of factorials in Lucas sequences
We prove that if {un}n≥ 0 is a nondegenerate Lucas sequence, then there are only finitely many effectively computable positive integers n such that |un|=f(m!), where f is either the sum-of-divisors function, or the sum-of-proper-divisors function, or the Euler phi function. We also give a theorem that holds for a more general class of integer sequences and illustrate our results through a few specific examples. This paper is motivated by a previous work of Iannucci and Luca who addressed the above problem with Catalan numbers and the sum-of-proper-divisors function
Mersenne k-Fibonacci numbers
For an integer k≥ 2, let (Fn(k))n be the k-Fibonacci sequence which starts with 0,...,0,1 (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all k-Fibonacci numbers which are Mersenne numbers, i.e., k-Fibonacci numbers that are equal to 1 less than a power of 2. As a consequence, for each fixed k, we prove that there is at most one Mersenne prime in (Fn(k))n
Fermat -Fibonacci and -Lucas numbers
summary:Using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and Pethő, we find all -Fibonacci and -Lucas numbers which are Fermat numbers. Some more general results are given
Ratios of sums of two Fibonacci numbers equal to powers of 2
In this paper, we find all solutions to the Diophantine equation , where is the Fibonacci sequence. This paper continues and extends a previous work which investigated the powers of 2 which are sums of two Fibonacci numbers