A note on the squares of the form $\prod_{k=1}^n (2k^2+l)$ with $l$ odd

Russelle Guadalupe

A note on the squares of the form $\prod_{k=1}^n (2k^2+l)$ with $l$ odd

Number Theory 2023-12-12 v2

Authors: Russelle Guadalupe

Abstract

Let $l$ be a positive odd integer. Using Cilleruelo's method, we establish an explicit lower bound $N_l$ depending on $l$ such that for all $n\geq N_l$ , $\prod_{k=1}^n (2k^2+l)$ is not a square. As an application, we determine all values of $n$ such that $\prod_{k=1}^n (2k^2+l)$ is a square for certain values of $l$ .

Keywords

squares product of quadratic forms odd integer cilleruelo's method lower bound number theory

Cite

@article{arxiv.2201.00501,
  title  = {A note on the squares of the form $\prod_{k=1}^n (2k^2+l)$ with $l$ odd},
  author = {Russelle Guadalupe},
  journal= {arXiv preprint arXiv:2201.00501},
  year   = {2023}
}

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