Fibonacci number

About: Fibonacci number is a research topic. Over the lifetime, 6711 publications have been published within this topic receiving 65977 citations.

Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers

Abstract: This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144 and the only perfect powers in the Lucas sequence are 1, 4.

Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian

Abstract: It is rigorously proven that the spectrum of the tight-binding Fibonacci Hamiltonian,H mn=δ m, n+1+δ m, n−1+δ m, n μ[(n+1)α]−[nα]) where α=(√5−1)/2 and [·] means integer part, is a Cantor set of zero Lebesgue measure for all real nonzeroμ, and the spectral measures are purely singular continuous. This follows from a recent result by Kotani, coupled with the vanishing of the Lyapunov exponent in the spectrum.

Universal Topological Quantum Computation from a Superconductor-Abelian Quantum Hall Heterostructure

Abstract: Non-Abelian anyons promise to reveal spectacular features of quantum mechanics that could ultimately provide the foundation for a decoherence-free quantum computer. A key breakthrough in the pursuit of these exotic particles originated from Read and Green’s observation that the Moore-Read quantum Hall state and a (relatively simple) two-dimensional p+ip superconductor both support so-called Ising non-Abelian anyons. Here, we establish a similar correspondence between the Z_3 Read-Rezayi quantum Hall state and a novel two-dimensional superconductor in which charge-2e Cooper pairs are built from fractionalized quasiparticles. In particular, both phases harbor Fibonacci anyons that—unlike Ising anyons—allow for universal topological quantum computation solely through braiding. Using a variant of Teo and Kane’s construction of non-Abelian phases from weakly coupled chains, we provide a blueprint for such a superconductor using Abelian quantum Hall states interlaced with an array of superconducting islands. Fibonacci anyons appear as neutral deconfined particles that lead to a twofold ground-state degeneracy on a torus. In contrast to a p+ip superconductor, vortices do not yield additional particle types, yet depending on nonuniversal energetics can serve as a trap for Fibonacci anyons. These results imply that one can, in principle, combine well-understood and widely available phases of matter to realize non-Abelian anyons with universal braid statistics. Numerous future directions are discussed, including speculations on alternative realizations with fewer experimental requirements.

Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers

Abstract: This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect powers in the Lucas sequence are 1 and 4.