Kantaphon Kuhapatanakul, Natnicha Meeboomak and Kanyarat Thongsing

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310β5132, Online ISSN 2367β8275

Volume 24, 2018, Number 3, Pages 56β61

DOI: 10.7546/nntdm.2018.24.3.56-61

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## Details

### Authors and affiliations

Kantaphon Kuhapatanakul

*Department of Mathematics, Faculty of Science,
Kasetsart University, Bangkok, Thailand*

Natnicha Meeboomak

*Department of Mathematics, Faculty of Science,
Kasetsart University, Bangkok, Thailand*

Kanyarat Thongsing

*Department of Mathematics, Faculty of Science,
Kasetsart University, Bangkok, Thailand*

### Abstract

Let π be a positive integer. We study the Diophantine equation Ξ οΈ_{k=1}^{n}(π^{2}π^{4} + (2π β π^{2})π^{2} + 1) = π¦^{2}. This Diophantine equation generalizes a result of GΓΌrel [5] for π = 2. We also prove that the product (2^{2} β 1)(3^{2} β 1)β¦(π^{2} β 1) is a perfect square only for the values π for which the triangular number T^{n} is a perfect square.

### Keywords

- Diophantine equation
- Perfect square
- Quartic polynomial
- Quadratic polynomial

### 2010 Mathematics Subject Classification

- 11D25
- 11D09

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## Cite this paper

Kuhapatanakul, K., Meeboomak , N., & Thongsing, K. (2018). On products of quartic polynomials over consecutive indices which are perfect squares. Notes on Number Theory and Discrete Mathematics, 24(3), 56-61, doi: 10.7546/nntdm.2018.24.3.56-61.