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

^*Corresponding author for this work

Institute of Analysis and Number Theory (5010)

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

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.

Original language	English
Pages (from-to)	651-684
Number of pages	34
Journal	The Ramanujan Journal
Volume	56
Early online date	23 Jul 2020
DOIs	https://doi.org/10.1007/s11139-020-00278-7
Publication status	Published - 2020

Keywords

Baker’s method
Linear forms in logarithms
Lucas sequences

ASJC Scopus subject areas

Algebra and Number Theory

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

https://doi.org/10.33774/coe-2020-d0vq5Licence: CC BY 4.0

Cite this

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title = "On the exponential Diophantine equation related to powers of two consecutive terms of Lucas sequences",

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",

author = "Mahadi Ddamulira and Florian Luca",

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doi = "10.1007/s11139-020-00278-7",

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AU - Ddamulira, Mahadi

AU - Luca, Florian

PY - 2020

Y1 - 2020

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

AN - SCOPUS:85088477705

SN - 1572-9303

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SP - 651

EP - 684

JO - The Ramanujan Journal

JF - The Ramanujan Journal

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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