Euclidean algorithm

The Euclidean algorithms for finding a compound ratio also allowed a ratio and an inverse ratio, and more than two ratios to be compounded, since each compounded pair is equivalent to a single ratio between lines.

In a quadratic field Q (√), D a squarefree integer, with class number 1 any algebraic integer can be decomposed uniquely into primes but for only 21 domains Euclidean algorithms are known. We prove that for D<-19 even remainder sequences with possibly nondecreasing norms cannot determine the GCD of arbitrary inputs.