Related papers: Zeckendorf arithmetic for Lucas numbers
Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail.…
Zeckendorf's theorem states that every positive integer can be written uniquely as the sum of non-consecutive shifted Fibonacci numbers $\{F_n\}$, where we take $F_1=1$ and $F_2=2$. This has been generalized for any Positive Linear…
We study partitions of totally positive integers in real quadratic fields. We develop an algorithm for computing the number of partitions, prove a result about the parity of the partition function, and characterize the quadratic fields such…
In this paper, we find all positive squarefree integers d such that the Pell equation X2-dY2 = +-1 has at least two positive integer solutions (X,Y) and (X',Y') such that both X and X' have Zeckendorf representations with at most two terms.…
We develop an algorithm for computing affine Kazhdan-Lusztig polynomials, for all Lie types. This generalizes our previously published algorithm for type A, which in turn is a faster version of an algorithm due to Lascouz, Leclerc and…
We study algebraic algorithms for expressing the number of non-negative integer solutions to a unimodular system of linear equations as a function of the right hand side. Our methods include Todd classes of toric varieties via Gr\"obner…
We use generalised Zeckendorf representations of natural numbers to investigate mixing properties of symbolic dynamical systems. The systems we consider consist of bi-infinite sequences associated with so-called random substitutions. We…
We give a number of theoretical and practical methods related to the computation of L-functions, both in the local case (counting points on varieties over finite fields, involving in particular a detailed study of Gauss and Jacobi sums),…
We give elementary proofs of some congruence criteria to compute binomial coefficients in modulo a prime. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas Theorem by using…
In this paper we classify the Zeckendorf expansions according to their digit blocks. It turns out that if we consider these digit blocks as labels on the Fibonacci tree, then the numbers ending with a given digit block in their Zeckendorf…
Recent developments of Baxter algebras have lead to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first kind and the second kind, partitions and multinomial…
We apply the geometric construction of solutions of some variational problems of combinatorics to estimate the number of partitions and of plane partitions of an integer.
Dual numbers and their higher order version are important tools for numerical computations, and in particular for finite difference calculus. Based upon the relevant algebraic rules and matrix realizations of dual numbers, we will present a…
Zeckendorf's theorem states every positive integer has a unique decomposition as a sum of non-adjacent Fibonacci numbers. This result has been generalized to many sequences $\{a_n\}$ arising from an integer positive linear recurrence, each…
We use recurrence equations (alias difference equations) to enumerate the number of formula-representations of positive integers using only addition and multiplication, and using addition, multiplication, and exponentiation, where all the…
As a variant of Zeckendorf's theorem, Chung and Graham proved that every positive integer can be uniquely decomposed into a sum of even-indexed Fibonacci numbers, whose coefficients are either $0, 1$, or $2$ so that between two coefficients…
Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-$b$ expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence.…
We consider the sum of digits functions for both base phi, and for the Zeckendorf expansion of the natural numbers. For both sum of digits functions we present morphisms on infinite alphabets such that these functions viewed as infinite…
Lucas' theorem describes how to reduce a binomial coefficient $\binom{a}{b}$ modulo $p$ by breaking off the least significant digits of $a$ and $b$ in base $p$. We characterize the pairs of these digits for which Lucas' theorem holds modulo…
The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In…