Projects

The Diophatine Equation $\displaystyle \prod_{k=1}^{n}\left(1+k^2\right)=b^2$ (Jun 2022 - Dec 2022)

This paper discusses integer solutions (in fact there is only one pair) to the Diophantine Equation \[\prod_{k=1}^{n}\left(1+k^2\right)=b^2.\] It heavily relies on techniques in Analytic Number Theory and we improve on a bound obtained by J. Cilleruelo. This problem was motivated by the Three Square Geometry Problem discussed on Numberphile, for which one can extend to asking whether there exists rational multiples of $\pi$ which can be written as \[\sum_{k=1}^{N}\operatorname{arctan}\left(\frac{1}{k}\right).\]