## Functiones et Approximatio Commentarii Mathematici

### Products of consecutive values of some quartic polynomials

Artūras Dubickas

#### 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

https://projecteuclid.org/euclid.facm/1540519331

Digital Object Identifier
doi:10.7169/facm/1733

Mathematical Reviews number (MathSciNet)
MR3964262

Zentralblatt MATH identifier
07068533

#### 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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