Functiones et Approximatio Commentarii Mathematici

Products of consecutive values of some quartic polynomials

Artūras Dubickas

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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

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

First available in Project Euclid: 26 October 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

integer polynomial Pell's equation perfect square


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.

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