Sum Of Fibonacci Numbers Research Articles

Sums of Fibonacci numbers that are perfect powers

Let us denote by Fn the n-th Fibonacci number. In this paper we show that for a fixed integer y there exists at most one integer exponent a > 0 such that the Diophantine equation Fn + Fm = ya has a solution (n, m, a) in positive integers satisfying n > m > 0, unless y = 2, 3, 4, 6 or 10.

Proof Without Words: Sum of Fibonacci Numbers and Beyond

Proof Without Words: Sum of Fibonacci Numbers and Beyond

On sums of Fibonacci numbers with few binary digits

In this paper we completely solve the Diophantine equation $F_n+F_m=2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$, where $F_k$ denotes the $k$-th Fibonacci number. In addition to complex linear forms in logarithms and the Baker-Davenport reduction method, we use $p$-adic versions of both tools.

Representation of Integers as Sums of Fibonacci and Lucas Numbers

Motivated by the Elementary Problem B-416 in the Fibonacci Quarterly, we show that, given any integers n and r with n≥2, every positive integer can be expressed as a sum of Fibonacci numbers whose indices are distinct integers not congruent to r modulo n. Similar expressions are also dealt with for the case of Lucas numbers. Symmetric and anti-symmetric properties of Fibonacci and Lucas numbers are used in the proofs.

Wythoff partizan subtraction

We introduce a class of normal-play partizan games, called Complementary Subtraction. These games are instances of Partizan Subtraction where we take any set A of positive integers to be Left’s subtraction set and let its complement be Right’s subtraction set. In wythoff partizan subtraction we take the set A and its complement B from wythoff nim, as the two subtraction sets. As a function of the heap size, the maximum size of the canonical forms grows quickly. However, the value of the heap is either a number or, in reduced canonical form, a switch. We find the switches by using properties of the Fibonacci word and standard Fibonacci representations of integers. Moreover, these switches are invariant under shifts by certain Fibonacci numbers. The values that are numbers, however, are distinct, and we can find their binary representation in polynomial time using a representation of integers as sums of Fibonacci numbers, known as the ternary (or “the even”) Fibonacci representation.

Melham's Conjecture on Odd Power Sums of Fibonacci Numbers

Ozeki and Prodinger showed that the odd power sum of the first several consecutive Fibonacci numbers of even order is equal to a polynomial evaluated at a certain Fibonacci number of odd order. We prove that this polynomial and its derivative both vanish at 1, and will be an integer polynomial after multiplying it by a product of the first consecutive Lucas numbers of odd order. This presents an affirmative answer to a conjecture of Melham.

Differences of Multiple Fibonacci Numbers

AbstractWe show that every integer can be written uniquely as a sum of Fibonacci numbers and their additive inverses, such that every two terms of the same sign differ in index by at least 4 and every two terms of different sign differ in index by at least 3. Furthermore, there is no way to use fewer terms to write a number as a sum of Fibonacci numbers and their additive inverses. This is an analogue of the Zeckendorf representation.

Sums of Fibonacci Numbers by Matrix Methods

Sums of Fibonacci Numbers by Matrix Methods

On Minimal Number of Terms in Representation of Natural Numbers as a Sum of Fibonacci Numbers

On Minimal Number of Terms in Representation of Natural Numbers as a Sum of Fibonacci Numbers

On Sums of Fibonacci Numbers

On Sums of Fibonacci Numbers