TY - JOUR

T1 - The weak converse of Zeckendorf’s theorem

AU - Chang, Sungkon

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2021/9

Y1 - 2021/9

N2 - By Zeckendorf’s Theorem, every positive integer is uniquely written as a sum of non-adjacent terms of the Fibonacci sequence, and its converse states that if a sequence in the positive integers has this property, it must be the Fibonacci sequence. If we instead consider the problem of finding a monotone sequence with such a property, we call it the weak converse of Zeckendorf’s theorem. In this paper, we first introduce a generalization of Zeckendorf conditions, and subsequently, Zeckendorf’s theorems and their weak converses for the general Zeckendorf conditions. We also extend the generalization and results to the real numbers in the interval (0, 1), and to p-adic integers.

AB - By Zeckendorf’s Theorem, every positive integer is uniquely written as a sum of non-adjacent terms of the Fibonacci sequence, and its converse states that if a sequence in the positive integers has this property, it must be the Fibonacci sequence. If we instead consider the problem of finding a monotone sequence with such a property, we call it the weak converse of Zeckendorf’s theorem. In this paper, we first introduce a generalization of Zeckendorf conditions, and subsequently, Zeckendorf’s theorems and their weak converses for the general Zeckendorf conditions. We also extend the generalization and results to the real numbers in the interval (0, 1), and to p-adic integers.

KW - Representation of numbers

KW - Zeckendorf’s theorem

UR - http://www.scopus.com/inward/record.url?scp=85111284690&partnerID=8YFLogxK

U2 - 10.1007/s40993-021-00275-9

DO - 10.1007/s40993-021-00275-9

M3 - Article

AN - SCOPUS:85111284690

SN - 2363-9555

VL - 7

JO - Research in Number Theory

JF - Research in Number Theory

IS - 3

M1 - 48

ER -