Let {Fn}n=1 = {1, 1, 2, 3, . . .} be the sequence of Fibonacci numbers. In this paper we give some sufficient conditions on a natural number k such that the equation Fn = kFm is solvable with respect to the unknowns n and m. We also show that for k > 1 the equation Fn = kFm has at most one solution (n, m).

In this paper we study the Diophantine equation x −6xy +5y = 16Fn−1Fn+1, where (Fn) n≥0 is the Fibonacci sequence and we find a class of such equations having solutions which are determined.

In this article, a novel and efficient collocation method based on Fibonacci wavelets is proposed for the numerical solution of non-linear Hunter–Saxton equation. Firstly, operational matrices integration associated with are constructed by following strategy Chen Hsiao. The merged used to convert given problem into system algebraic equations that can be solved any classical method, such as Newt...

The m-th Cullen number Cm is a number of the form m2 m + 1 and the m-th Woodall number Wm has the form m2 m − 1. In 2003, Luca and Stănică proved that the largest Fibonacci number in the Cullen sequence is F4 = 3 and that F1 = F2 = 1 are the largest Fibonacci numbers in the Woodall sequence. Very recently, the second author proved that, for any given s > 1, the equation Fn = ms m ± 1 has only f...

Low energy properties of the S=1/2 antiferromagnetic Heisenberg chains with Fibonacci exchange modulation are studied using the real space renormalization group method for strong exchange modulation. Using the analytical solution of the recursion equation, the true asymptotic behavoir is revealed, which was veiled by the finite size effect in the previous numerical works. It is found that the g...

We study the electronic density of states (DOS) of a GaAs/GaxAl1 xAs superlattice with impurities arranged according to the Fibonacci sequence. Our theory uses a Green function method based on Dyson’s equation within the real-space renormalizationgroup approach in the framework of the tight-binding Hamiltonian for the Fibonacci quasiperiodic lattice. An exact decimation transformation has been ...

Let z(N) be the order of appearance of N in the Fibonacci sequence. This is the smallest positive integer k such that N divides the kth Fibonacci number. We show that each of the 6 total possible orderings among z(N), z(N + 1), z(N + 2) appears infinitely often. We also show that for each nonzero even integer c and many odd integers c the equation z(N) = z(N + c) has infinitely many solutions N...

We study the electronic structure of a new type of Fibonacci superlattice based on Si δ-doped GaAs. Assuming that δ-doped layers are equally spaced, quasiperiodicity is introduced by selecting two different donor concentrations and arranging them according to the Fibonacci series along the growth direction. The one-electron potential due to δ-doping is obtained by means of the Thomas-Fermi appr...