The number of steps in the Euclidean algorithm over complex quadratic fields (Q807674)

Let \(L(u,v)=N\) be the least possible length of the number of divisions in the Euclidean algorithm of any two integers in \({\mathbb{Q}}(\sqrt{-m})\) where \(m=1,2,3,7\) or 11. The following result is proved: Let \(x\geq | v| \geq | u| >0\). The proportion of pairs for which \(c_ 1(m) \ln x\leq L(u,v)\leq c_ 2(m) \ln x\) tends to 1 as \(x\to \infty\), where \(c_ 1(m)\) and \(c_ 2(m)\) are positive constants depending only on m.