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June 2019 Products of consecutive values of some quartic polynomials
Artūras Dubickas
Funct. Approx. Comment. Math. 60(2): 237-244 (June 2019). DOI: 10.7169/facm/1733

Abstract

In this paper, we investigate some special quartic polynomials $P$ whose coefficients for $x^4,x^3,\dots,1$ are $a^2, 2a(a+b), a^2+b^2+3ab+2ac, (a+b)(b+2c), (a+b+c)c$, where $a,b,c \in \mathbb{Z}$, and consider the question whether the product $\prod_{k=1}^m P(k)$ is a perfect square for infinitely many $m \in \mathbb{N}$ or for only finitely many $m \in \mathbb{N}$. The answer depends on the solutions of the Pell type diophantine equation $(a+b+c)(ax^2+bx+c)=y^2$. Our results imply, for example, that the product $\prod_{k=1}^m (4k^4+8k^2+9)$ is a perfect square for infinitely many $m \in \mathbb{N}$, whereas the product $\prod_{k=1}^m (k^4+7k^2+16)$ is a perfect square for $m=3$ only, when it equals $230400=480^2$.

Citation

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Artūras Dubickas. "Products of consecutive values of some quartic polynomials." Funct. Approx. Comment. Math. 60 (2) 237 - 244, June 2019. https://doi.org/10.7169/facm/1733

Information

Published: June 2019
First available in Project Euclid: 26 October 2018

zbMATH: 07068533
MathSciNet: MR3964262
Digital Object Identifier: 10.7169/facm/1733

Subjects:
Primary: 11D09
Secondary: 11B83 , 11D45

Keywords: integer polynomial , Pell's equation , perfect square

Rights: Copyright © 2019 Adam Mickiewicz University

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Vol.60 • No. 2 • June 2019
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