Arithmetics in Other Quadratic Rings

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 \(2^{n+2}\)).