Every quadratic ring belongs to one of the two classes:

• $$1^{\circ}$$ Extensions of the form $$K=\mathbb{Z}[\sqrt{d}]$$, where $$d\neq1$$ is a squarefree integer. The conjugation and norm are given by the formulas $$\overline{x+y\sqrt{d}}=x-y\sqrt{d}$$ and $$N(x+y\sqrt{d})=x^2-dy^2$$, where $$x,y\in\mathbb{Z}$$.

• $$2^{\circ}$$ Extensions of the form $$K=\mathbb{Z}[\alpha]$$ for $$\alpha=\frac{-1+\sqrt{d}}2$$, where $$d=4k+1$$ ($$k\in\mathbb{Z}$$) is a squarefree integer with $$d\neq1$$ (then $$\alpha$$ is an algebraic integer: $$\alpha^2+\alpha-k=0$$). The conjugation and norm are given by $$\overline{x+y\alpha}= x-y-y\alpha$$ and $$N(x+y\alpha)=x^2-xy-ky^2$$, where $$x,y\in\mathbb{Z}$$.

Some of these rings are Euclidean, such as $$\mathbb{Z}[\sqrt{d}]$$ for $$d=-2,-1,2,3,6,7$$ and $$\mathbb{Z}\left[\frac{-1+\sqrt{d}}2 \right]$$ for $$d=-7,-3,5$$.

Determining all quadratic unique factorization rings (including the non-Euclidean ones) is extremely serious. Among the rings of the type $$1^{\circ}$$ and $$2^{\circ}$$ with $$d< 0$$, apart from the ones mentioned already, the FTA holds in only five other rings: namely, the rings of the type $$2^{\circ}$$ for $$d=-11,-19,-43,-67,-163$$. Gauss’ conjecture that the FTA holds in infinitely many quadratic rings with a positive $$d$$ has not been proved nor disproved until today.

Problem 6

Find all integer solutions of the equation $$x^2+2=y^3$$.
Problem 7

Consider the sequence $$a_0,a_1,a_2,\dots$$ given by $$a_0=2$$ and $$a_{k+1}=2a_k^2-1$$ for $$k\geq0$$. Prove that if an odd prime number $$p$$ divides $$a_n$$, then $$p\equiv\pm1$$ (mod $$2^{n+2}$$).

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