On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences

Mahadi Ddamulira; Florian Luca

doi:10.1007/s11139-020-00278-7

On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences

Mahadi Ddamulira^*, Florian Luca

^*Korrespondierende/r Autor/-in für diese Arbeit

Institut für Analysis und Zahlentheorie (5010)

Publikation: Beitrag in einer Fachzeitschrift › Artikel › Begutachtung

Abstract

Let r≥ 1 be an integer and U:=(Un)n≥0 be the Lucas sequence given by U= 0 , U₁= 1 , and U_n₊₂= rU_n₊₁+ U_n, for all n≥ 0. In this paper, we show that there are no positive integers r≥3,x≠2,n≥1 such that Unx+Un+1x is a member of U.

Originalsprache	englisch
Seiten (von - bis)	651-684
Seitenumfang	34
Fachzeitschrift	The Ramanujan Journal
Jahrgang	56
Frühes Online-Datum	23 Juli 2020
DOIs	https://doi.org/10.1007/s11139-020-00278-7
Publikationsstatus	Veröffentlicht - 2020

ASJC Scopus subject areas

Algebra und Zahlentheorie

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10.1007/s11139-020-00278-7Lizenz: CC BY 4.0

https://doi.org/10.33774/coe-2020-d0vq5Lizenz: CC BY 4.0

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http://www.scopus.com/inward/record.url?scp=85088477705&partnerID=8YFLogxK

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abstract = "Let r≥ 1 be an integer and U:=(Un)n≥0 be the Lucas sequence given by U= 0 , U1= 1 , and Un+2= rUn+1+ Un, for all n≥ 0. In this paper, we show that there are no positive integers r≥3,x≠2,n≥1 such that Unx+Un+1x is a member of U.",

keywords = "Baker{\textquoteright}s method, Linear forms in logarithms, Lucas sequences",

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AU - Ddamulira, Mahadi

AU - Luca, Florian

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N2 - Let r≥ 1 be an integer and U:=(Un)n≥0 be the Lucas sequence given by U= 0 , U1= 1 , and Un+2= rUn+1+ Un, for all n≥ 0. In this paper, we show that there are no positive integers r≥3,x≠2,n≥1 such that Unx+Un+1x is a member of U.

AB - Let r≥ 1 be an integer and U:=(Un)n≥0 be the Lucas sequence given by U= 0 , U1= 1 , and Un+2= rUn+1+ Un, for all n≥ 0. In this paper, we show that there are no positive integers r≥3,x≠2,n≥1 such that Unx+Un+1x is a member of U.

KW - Baker’s method

KW - Linear forms in logarithms

KW - Lucas sequences

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U2 - 10.1007/s11139-020-00278-7

DO - 10.1007/s11139-020-00278-7

M3 - Article

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JO - The Ramanujan Journal

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On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences

Abstract

ASJC Scopus subject areas

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