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arXiv · 2008.10718

Four conjectures by Zhi-Hong Sun

Constantin N. Beli
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Abstract

We prove some results conjectured by Zhi-Hong Sun regarding the value $\mod p$ of $\varepsilon_d^{\frac{p-1}4}$, where $\varepsilon_d$ is a unit of norm $-1$ in some fields $\mathbb Q(\sqrt d)$, with $\left(\frac{-1}p\right) =\left(\frac dp\right) =1$. The answer is given in terms of how $p$ writes as $p=f(x,y)=u^2+v^2$, with $x,y,u,v\in\mathbb Z$, where $f$ is a certain quadratic form of determinant $-4d$.

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Constantin N. Beli. 2020-08-24. Four conjectures by Zhi-Hong Sun. https://arxiv.org/abs/2008.10718

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