Lucas sequences whose 12th or 9th term is a square

Andrew Bremner; N. Tzanakis

doi:10.1016/j.jnt.2004.04.002

Lucas sequences whose 12th or 9th term is a square

Andrew Bremner, N. Tzanakis

Mathematical and Statistical Sciences, School of (SoMSS)

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

14 Scopus citations

Abstract

Let P and Q be non-zero relatively prime integers. The Lucas sequence {U_n(P,Q)} is defined by U₀=0, U₁=1, U_n=PU_n-1-QU_n-2 (n≥2). We show that the only sequence with U₁₂(P,Q) a perfect square is the Fibonacci sequence {U_n(1,-1)}; and we show that there are no non-degenerate sequences {U_n(P,Q)} with U₉(P,Q) a perfect square. The argument involves finding all rational points on several curves of genus 2.

Original language	English (US)
Pages (from-to)	215-227
Number of pages	13
Journal	Journal of Number Theory
Volume	107
Issue number	2
DOIs	https://doi.org/10.1016/j.jnt.2004.04.002
State	Published - Aug 2004

Keywords

Elliptic curve
Formal group
Lucas sequence

ASJC Scopus subject areas

Algebra and Number Theory

Access to Document

10.1016/j.jnt.2004.04.002

Cite this

@article{713e2f12991f4e9e982658ba73e10541,

title = "Lucas sequences whose 12th or 9th term is a square",

abstract = "Let P and Q be non-zero relatively prime integers. The Lucas sequence {Un(P,Q)} is defined by U0=0, U1=1, Un=PUn-1-QUn-2 (n≥2). We show that the only sequence with U12(P,Q) a perfect square is the Fibonacci sequence {Un(1,-1)}; and we show that there are no non-degenerate sequences {Un(P,Q)} with U9(P,Q) a perfect square. The argument involves finding all rational points on several curves of genus 2.",

keywords = "Elliptic curve, Formal group, Lucas sequence",

author = "Andrew Bremner and N. Tzanakis",

year = "2004",

month = aug,

doi = "10.1016/j.jnt.2004.04.002",

language = "English (US)",

volume = "107",

pages = "215--227",

journal = "Journal of Number Theory",

issn = "0022-314X",

publisher = "Academic Press Inc.",

number = "2",

}

TY - JOUR

T1 - Lucas sequences whose 12th or 9th term is a square

AU - Bremner, Andrew

AU - Tzanakis, N.

PY - 2004/8

Y1 - 2004/8

N2 - Let P and Q be non-zero relatively prime integers. The Lucas sequence {Un(P,Q)} is defined by U0=0, U1=1, Un=PUn-1-QUn-2 (n≥2). We show that the only sequence with U12(P,Q) a perfect square is the Fibonacci sequence {Un(1,-1)}; and we show that there are no non-degenerate sequences {Un(P,Q)} with U9(P,Q) a perfect square. The argument involves finding all rational points on several curves of genus 2.

AB - Let P and Q be non-zero relatively prime integers. The Lucas sequence {Un(P,Q)} is defined by U0=0, U1=1, Un=PUn-1-QUn-2 (n≥2). We show that the only sequence with U12(P,Q) a perfect square is the Fibonacci sequence {Un(1,-1)}; and we show that there are no non-degenerate sequences {Un(P,Q)} with U9(P,Q) a perfect square. The argument involves finding all rational points on several curves of genus 2.

KW - Elliptic curve

KW - Formal group

KW - Lucas sequence

UR - http://www.scopus.com/inward/record.url?scp=4344625641&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4344625641&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2004.04.002

DO - 10.1016/j.jnt.2004.04.002

M3 - Article

AN - SCOPUS:4344625641

SN - 0022-314X

VL - 107

SP - 215

EP - 227

JO - Journal of Number Theory

JF - Journal of Number Theory

IS - 2

ER -

Lucas sequences whose 12th or 9th term is a square

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this