Some Identities Involving Square of Fibonacci Numbers and Lucas Numbers

By studying the properties of Chebyshev polynomials, some specific and mean-ingful identities for the calculation of square of Chebyshev polynomials, Fibonacci numbersand Lucas numbers are obtained.

On the Norms of r-Toeplitz Matrices Involving Fibonacci and Lucas Numbers

Let us define to be a r-Toeplitz matrix. The entries in the first row of are or;where Fn and Ln denote the usual Fibonacci and Lucas numbers, respectively. We obtained some bounds for the spectral norm of these matrices.

Generalizations of Some Formulas of the Reciprocal Sum and Alternating Sum for Generalized Lucas Numbers

The purpose of this article is to provide the inversion relationships between the reciprocal sum S（1, 2,…, m） and the alternating sum T（1, 2,…, m） for generalized Lucas numbers which generalizes the Melham＇s results.

On the Norms of r-Hankel Matrices Involving Fibonacci and Lucas Numbers

Let us define A=Hr=(aij)?to be n×n?r-Hankel matrix. The entries of matrix A are Fn=Fi+j-2?or Ln=Fi+j-2?where Fn?and Ln?denote the usual Fibonacci and Lucas numbers, respectively. Then, we obtained upper and lower bounds for the spectral norm of matrix A. We compared our bounds with exact value of matrix A’s spectral norm. These kinds of matrices have connections with signal and image processing, time series analysis and many other problems.

Some Sum Formulas of ( s , t )-Jacobsthal and ( s , t )-Jacobsthal Lucas Matrix Sequences

In this study, we first give the definitions of (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas sequence. By using these formulas we define (s,t)-Jacobsthal and (s,t)-Jacobsthal Lucas matrix sequences. After that we establish some sum formulas for these matrix sequences.

Golden Quartic Polynomial and Moebius-Ball Electron

A symmetrical quartic polynomial, named golden one, can be connected to coefficients of the icosahedron equation as well as to the gyromagnetic correction of the electron and to number 137. This number is not a mystic one, but is connected with the inverse of Sommerfeld’s fine-structure constant and this way again connected with the electron. From number-theoretical realities, including the reciprocity relation of the golden ratio as effective pre-calculator of nature’s creativeness, a proposed closeness to the icosahedron may point towards the structure of the electron, thought off as a single-strand compacted helically self-confined charged elemantary particle of less spherical but assumed blunted icosahedral shape generated from a high energy double-helix photon. We constructed a chiral Moebius “ball” from a 13 times 180˚twisted double helix strand, where the turning points of 12 generated slings were arranged towards the vertices of a regular icosahedron, belonging to the non-centrosymmetric rotation group I532. Mathematically put, we convert the helical motion of an energy quantum into a stationary motion on a Moebius stripe structure. The radius of the ball is about the Compton radius. This chiral closed circuit Moebius ball motion profile can be tentatively thought off as the dominant quantum vortex structure of the electron, and the model may be named CEWMB (Charged Electromagnetic Wave Moebius Ball). Also the gyromagnetic factor of the electron (ge = 2.002319) can be traced back to this special structure. However, nature’s energy infinity principle would suggest a superposition with additional less dominant (secondary) structures, governed also by the golden mean. A suggestion about the possible structure of delocalized hole carriers in the superconducting state is given.

Squares from D(-4)and D(20)Triples

We study the eight infinite sequences of triples of natural numbers A=(F2n+1,4F2n+3,F2n+7), B=(F2n+1,4F2n+5,F2n+7), C=(F2n+1,5F2n+1,F2n+3), D=(F2n+3,4F2n+1,F2n+3) and A=(L2n+1,4L2n+3,L2n+7), B=(L2n+1,4L2n+5,L2n+7), C=(L2n+1,5L2n+1,L2n+3), D=(L2n+3,4L2n+1,L2n+3. The sequences A,B,C and D are built from the Fibonacci numbers Fn while the sequences A, B, C and D from the Lucas numbers Ln. Each triple in the sequences A,B,C and D has the property D(-4) (i. e., adding -4 to the product of any two different components of them is a square). Similarly, each triple in the sequences A, B, C and D has the property D(20). We show some interesting properties of these sequences that give various methods how to get squares from them.

Fibonacci and Lucas Congruences and Their Applications

In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some congruences concerning Fibonacci and Lucas numbers such as L2mn＋k ≡（-1）（m＋1）nLk（modLm）, F2mn＋k ≡（-1）（m＋1）nFk （modLm）, L2mn＋k ≡ （-1）mn Lk（mod Fm） and F2mn＋k≡ （-1）mn Fk （mod Fm）. By the achieved identities, divisibility properties of Fibonacci and Lueas numbers are given. Then it is proved that there is no Lucas number Ln such that Ln = L2ktLmx2 for m 〉 1 and k≥1. Moreover it is proved that Ln = LmLr is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given.

A Note on the Integrity of Certain Series

In this paper, using the theory of Pell equation, the authors discuss the integrity of certain series related to generalized Lucas numbers. Under some conditions, the integrity of certain series involving generalized Lucas numbers is completely solved.