Functiones et Approximatio Commentarii Mathematici

Products of consecutive values of some quartic polynomials

Artūras Dubickas

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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 $230400=480^2$.

Article information

Source
Funct. Approx. Comment. Math., Volume 60, Number 2 (2019), 237-244.

Dates
First available in Project Euclid: 26 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.facm/1540519331

Digital Object Identifier
doi:10.7169/facm/1733

Mathematical Reviews number (MathSciNet)
MR3964262

Zentralblatt MATH identifier
07068533

Subjects
Primary: 11D09: Quadratic and bilinear equations
Secondary: 11D45: Counting solutions of Diophantine equations 11B83: Special sequences and polynomials

Keywords
integer polynomial Pell's equation perfect square

Citation

Dubickas, Artūras. Products of consecutive values of some quartic polynomials. Funct. Approx. Comment. Math. 60 (2019), no. 2, 237--244. doi:10.7169/facm/1733. https://projecteuclid.org/euclid.facm/1540519331


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