4 problems

For squarefree $D>1$, let $\mathcal{O}$ be the ring of integers in $\mathbb{Q}(\sqrt{D})$, and let $\mathcal{S}_D$ consist of the elements $u=A+B\sqrt{D}$ with $A,B\in\mathbb{Z}$ s…

Let $p$) be a prime. The integer $p$ has the form … for integers $u$ and $v$, and a cubic residue modulo $p$ is an integer congruent modulo $p$ to a cube. Euler's conjecture. A pri…

Let $p$ be prime, let $m$ be a positive integer, and write $\mathbb{Z}_{p^m}$ for the integers modulo $p^m$. Let $CR$ denote a cubic residue and let $[i]$ denote the $i$th class in…

Let $p$ be a prime. The condition that $p$ has the form $u^2+27v^2$ is related to the congruence $p\equiv 1\pmod 3$ and cubic residuacity of $2$ modulo $p$. Euler's conjecture. The…