Representation of integers as sums of fibonacci and lucas numbers

Ho Park; Bumkyu Cho; Durkbin Cho; Yung Duk Cho; Joonsang Park

doi:10.3390/sym12101625

Representation of integers as sums of fibonacci and lucas numbers

Ho Park
, Bumkyu Cho
, Durkbin Cho
, Yung Duk Cho
, Joonsang Park

Department of Mathematics

Dongguk University

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

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.

Original language	English
Article number	1625
Pages (from-to)	1-8
Number of pages	8
Journal	Symmetry
Volume	12
Issue number	10
DOIs	https://doi.org/10.3390/sym12101625
State	Published - Oct 2020

Keywords

Fibonacci numbers
Lucas numbers
Zeckendorf’s theorem

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10.3390/sym12101625

Cite this

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abstract = "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.",

keywords = "Fibonacci numbers, Lucas numbers, Zeckendorf{\textquoteright}s theorem",

author = "Ho Park and Bumkyu Cho and Durkbin Cho and Cho, \{Yung Duk\} and Joonsang Park",

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AU - Park, Joonsang

PY - 2020/10

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N2 - 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.

AB - 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.

KW - Fibonacci numbers

KW - Lucas numbers

KW - Zeckendorf’s theorem

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DO - 10.3390/sym12101625

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ER -

Representation of integers as sums of fibonacci and lucas numbers

Abstract

Keywords

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