Arithmetic in Extensions of \(\mathbb Q\) (Table of contents)

Arithmetics in Other Quadratic Rings

Every quadratic ring belongs to one of the two classes:

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