Polynomial values in fibonacci sequences

Adi Ostrov; Danny Neftin; Avi Berman; Reyad A. Elrazik

doi:10.2140/involve.2020.13.597

Polynomial values in fibonacci sequences

Adi Ostrov, Danny Neftin, Avi Berman, Reyad A. Elrazik

Technion - Israel Institute of Technology

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

Abstract

The only perfect powers in the Fibonacci sequence are 0, 1, 8, and 144, and in the Lucas sequence, the only perfect powers are 1 and 4. We prove that in sequences that follow the same recurrence relation of the Lucas and Fibonacci sequences, there are always only finitely many polynomial values g(ℤ) for any polynomial g which is not equivalent to a Dickson polynomial.

Original language	English
Pages (from-to)	597-605
Number of pages	9
Journal	Involve
Volume	13
Issue number	4
DOIs	https://doi.org/10.2140/involve.2020.13.597
State	Published - 2020

Keywords

Fibonacci
Polynomial values
Recurrence sequence

All Science Journal Classification (ASJC) codes

General Mathematics

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10.2140/involve.2020.13.597

Cite this

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title = "Polynomial values in fibonacci sequences",

abstract = "The only perfect powers in the Fibonacci sequence are 0, 1, 8, and 144, and in the Lucas sequence, the only perfect powers are 1 and 4. We prove that in sequences that follow the same recurrence relation of the Lucas and Fibonacci sequences, there are always only finitely many polynomial values g(ℤ) for any polynomial g which is not equivalent to a Dickson polynomial.",

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AU - Ostrov, Adi

AU - Neftin, Danny

AU - Berman, Avi

AU - Elrazik, Reyad A.

PY - 2020

Y1 - 2020

N2 - The only perfect powers in the Fibonacci sequence are 0, 1, 8, and 144, and in the Lucas sequence, the only perfect powers are 1 and 4. We prove that in sequences that follow the same recurrence relation of the Lucas and Fibonacci sequences, there are always only finitely many polynomial values g(ℤ) for any polynomial g which is not equivalent to a Dickson polynomial.

AB - The only perfect powers in the Fibonacci sequence are 0, 1, 8, and 144, and in the Lucas sequence, the only perfect powers are 1 and 4. We prove that in sequences that follow the same recurrence relation of the Lucas and Fibonacci sequences, there are always only finitely many polynomial values g(ℤ) for any polynomial g which is not equivalent to a Dickson polynomial.

KW - Fibonacci

KW - Polynomial values

KW - Recurrence sequence

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U2 - 10.2140/involve.2020.13.597

DO - 10.2140/involve.2020.13.597

M3 - مقالة

SN - 1944-4176

VL - 13

SP - 597

EP - 605

JO - Involve

JF - Involve

IS - 4

ER -

Polynomial values in fibonacci sequences

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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