On products of quartic polynomials over consecutive indices which are perfect squares

Kantaphon Kuhapatanakul, Natnicha Meeboomak and Kanyarat Thongsing
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 3, Pages 56—61
DOI: 10.7546/nntdm.2018.24.3.56-61
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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 𝑎 be a positive integer. We study the Diophantine equation Π︁_k=1ⁿ(𝑎²𝑘⁴ + (2𝑎 − 𝑎²)𝑘² + 1) = 𝑦². This Diophantine equation generalizes a result of Gürel [5] for 𝑎 = 2. We also prove that the product (2² − 1)(3² − 1)…(𝑛² − 1) is a perfect square only for the values 𝑛 for which the triangular number Tⁿ is a perfect square.

Keywords

• Diophantine equation
• Perfect square
• Quartic polynomial
• Quadratic polynomial

2010 Mathematics Subject Classification

• 11D25
• 11D09

References

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