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


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