On squares in Lucas sequences

Andrew Bremner; N. Tzanakis

doi:10.1016/j.jnt.2006.10.007

On squares in Lucas sequences

Andrew Bremner, N. Tzanakis

Mathematical and Statistical Sciences, School of (SoMSS)

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

12 Scopus citations

Abstract

Let P and Q be non-zero integers. The Lucas sequence {U_n (P, Q)} is defined by U₀ = 0, U₁ = 1, U_n = P U_{n - 1} - Q U_{n - 2} (n ≥ 2). The question of when U_n (P, Q) can be a perfect square has generated interest in the literature. We show that for n = 2, ..., 7, U_n is a square for infinitely many pairs (P, Q) with gcd (P, Q) = 1; further, for n = 8, ..., 12, the only non-degenerate sequences where gcd (P, Q) = 1 and U_n (P, Q) = □, are given by U₈ (1, - 4) = 21², U₈ (4, - 17) = 620², and U₁₂ (1, - 1) = 12².

Original language	English (US)
Pages (from-to)	511-520
Number of pages	10
Journal	Journal of Number Theory
Volume	124
Issue number	2
DOIs	https://doi.org/10.1016/j.jnt.2006.10.007
State	Published - Jun 2007

Keywords

Genus two curves
Lucas sequence
Squares

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jnt.2006.10.007

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title = "On squares in Lucas sequences",

abstract = "Let P and Q be non-zero integers. The Lucas sequence {Un (P, Q)} is defined by U0 = 0, U1 = 1, Un = P Un - 1 - Q Un - 2 (n ≥ 2). The question of when Un (P, Q) can be a perfect square has generated interest in the literature. We show that for n = 2, ..., 7, Un is a square for infinitely many pairs (P, Q) with gcd (P, Q) = 1; further, for n = 8, ..., 12, the only non-degenerate sequences where gcd (P, Q) = 1 and Un (P, Q) = □, are given by U8 (1, - 4) = 212, U8 (4, - 17) = 6202, and U12 (1, - 1) = 122.",

keywords = "Genus two curves, Lucas sequence, Squares",

author = "Andrew Bremner and N. Tzanakis",

year = "2007",

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pages = "511--520",

journal = "Journal of Number Theory",

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TY - JOUR

T1 - On squares in Lucas sequences

AU - Bremner, Andrew

AU - Tzanakis, N.

PY - 2007/6

Y1 - 2007/6

N2 - Let P and Q be non-zero integers. The Lucas sequence {Un (P, Q)} is defined by U0 = 0, U1 = 1, Un = P Un - 1 - Q Un - 2 (n ≥ 2). The question of when Un (P, Q) can be a perfect square has generated interest in the literature. We show that for n = 2, ..., 7, Un is a square for infinitely many pairs (P, Q) with gcd (P, Q) = 1; further, for n = 8, ..., 12, the only non-degenerate sequences where gcd (P, Q) = 1 and Un (P, Q) = □, are given by U8 (1, - 4) = 212, U8 (4, - 17) = 6202, and U12 (1, - 1) = 122.

AB - Let P and Q be non-zero integers. The Lucas sequence {Un (P, Q)} is defined by U0 = 0, U1 = 1, Un = P Un - 1 - Q Un - 2 (n ≥ 2). The question of when Un (P, Q) can be a perfect square has generated interest in the literature. We show that for n = 2, ..., 7, Un is a square for infinitely many pairs (P, Q) with gcd (P, Q) = 1; further, for n = 8, ..., 12, the only non-degenerate sequences where gcd (P, Q) = 1 and Un (P, Q) = □, are given by U8 (1, - 4) = 212, U8 (4, - 17) = 6202, and U12 (1, - 1) = 122.

KW - Genus two curves

KW - Lucas sequence

KW - Squares

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On squares in Lucas sequences

Abstract

Keywords

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