1,114 research outputs found
On the greatest common divisor of and the th Fibonacci number
Let be the set of all integers of the form ,
where is a positive integer and denotes the th Fibonacci number.
We prove that for all
, and that has zero asymptotic density. Our proofs rely
on a recent result of Cubre and Rouse which gives, for each positive integer
, an explicit formula for the density of primes such that divides
the rank of appearance of , that is, the smallest positive integer such
that divides
Conway polynomials of two-bridge links
We give necessary conditions for a polynomial to be the Conway polynomial of
a two-bridge link. As a consequence, we obtain simple proofs of the classical
theorems of Murasugi and Hartley. We give a modulo 2 congruence for links,
which implies the classical modulo 2 Murasugi congruence for knots. We also
give sharp bounds for the coefficients of the Conway and Alexander polynomials
of a two-bridge link. These bounds improve and generalize those of Nakanishi
and Suketa.Comment: 15
SOME CONNECTIONS BETWEEN THE SMARANDACHE FUNCTION AND THE FIBONACCI SEQUENCE
This paper is aimed to provide generalizations of the Smarandache function. They
will be constructed by means of sequences more general than the sequence of the
factorials. Such sequences are monotonously convergent to zero sequences and divisibility sequences (in particular the Fibonacci sequence)
On the Fürstenberg closure of a class of binary recurrences
In this paper, we determine the closure in the full topology over Z of the set {un: n≥0}, where (un)n≥0 is a nondegenerate binary recurrent sequence with integer coefficients whose characteristic roots are quadratic units. This generalizes the result for the case when un=Fn was the nth Fibonacci number
On perfect powers that are sums of two Fibonacci numbers
We study the equation , where and are
respectively the -th and -th Fibonacci numbers and . We find all
solutions under the assumption .Comment: 6 page
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