A prime p is called a Fibonacci-Wieferich prime if Fp−( p5 ) ≡ 0 (mod p2), where Fn is the nth Fibonacci number. We report that there exist no such primes p < 2 × 1014. A prime p is called a Wolstenholme prime if (2p−1 p−1 ) ≡ 1 (mod p4). To date the only known Wolstenholme primes are 16843 and 2124679. We report that there exist no new Wolstenholme primes p < 109. Wolstenholme, in 1862, proved...

A numerical study is presented to investigate the electronic transport properties through a synthetic DNA molecule based on a quasiperiodic arrangement of its constituent nucleotides. Using a generalized Green's function technique, the electronic conduction through the poly(GACT)-poly(CTGA) DNA molecule in a metal/DNA/metal model structure has been studied. Making use of a renormalization schem...

The Fibonacci number of a graph is the number of independent vertex subsets. In this paper, we investigate trees with large Fibonacci number. In particular, we show that all trees with n edges and Fibonacci number > 2n−1 + 5 have diameter ≤ 4 and determine the order of these trees with respect to their Fibonacci numbers. Furthermore, it is shown that the average Fibonacci number of a star-like ...

We extend the notion of k-ribbon tableaux to the Fibonacci lattice, a differential poset defined by R. Stanley in 1975. Using this notion, we describe an insertion algorithm that takes k-colored permutations to pairs of k-ribbon Fibonacci tableaux of the same shape, and we demonstrate a colorto-spin property, similar to that described by Shimozono and White for ribbon tableaux. We give an evacu...

In this paper we focus on the combinatorial properties of the Fibonacci strings rotations. We first present a simple formula that, in constant time, determines the rank of any rotation (of a given Fibonacci string) in the lexicographically-sorted list of all rotations. We then use this information to deduce, also in constant time, the character that is stored at any one location of any given Fi...

This paper presents a newmethod to balance binary search trees, which has the following properties. (i) The only information stored for the balance is the size of every subtree. (ii) Inserting or deleting an element can be done in themost traditional way: first, the element is recursively inserted (deleted) in (from) the appropriate subtree; afterwards, a single or double rotation takes place i...

We consider the set partition statistics ls and rb introduced by Wachs and White and investigate their distribution over set partitions avoiding certain patterns. In particular, we consider those set partitions avoiding the pattern 13/2, Πn(13/2), and those avoiding both 13/2 and 123, Πn(13/2, 123). We show that the distribution over Πn(13/2) enumerates certain integer partitions, and the distr...

The aim of this paper is to give new results about factorizations of the Fibonacci numbers Fn and the Lucas numbers Ln. These numbers are defined by the second order recurrence relation an+2 = an+1+an with the initial terms F0 = 0, F1 = 1 and L0 = 2, L1 = 1, respectively. Proofs of theorems are done with the help of connections between determinants of tridiagonal matrices and the Fibonacci and ...

In a recent paper, Goyt and Sagan studied distributions of certain set partition statistics over pattern restricted sets of set partitions that were counted by the Fibonacci numbers. Their study produced a class of q-Fibonacci numbers, which they related to q-Fibonacci numbers studied by Carlitz and Cigler. In this paper we will study the distributions of some Mahonian statistics over pattern r...

Among numerical sequences, the Fibonacci numbers have achieved a kind of celebrity status with such fabulous properties, it is no wonder that the Fibonacci numbers stand out as a kind of super sequence. Fibonacci numbers have been studied in many different forms for centuries and the literature on the subject is consequently, incredibly vast. The Fibonacci sequence has been generalized in a num...