NUMBER SYSTEMS PERTAINING TO EUCLIDEAN RINGS OF IMAGINARY QUADRATIC INTEGERS

Abstract

Abstract. For a ring R of imaginary quadratic integers, using a conceptof a unitary number system in place of the Motzkin’s universal side divisor,we show that the following statements are equivalent:(1) R is Euclidean.(2) R has a unitary number system.(3) R is norm-Euclidean.Through an application of the above theorem we see that R admits binaryor ternary number systems if and only if R is Euclidean. 1. IntroductionIt is well known that among rings of imaginary quadratic integers, only nineringsZp 1; Zp 2; Z1 +p 32; Z1 + 72; Z1 +p 112;Z1 +p 192; Z1 +p 432; Z1 +p 672; Z1 +p 1632are principal ideal domains, which was conjectured by Gauss and settled com-pletely by Stark [15]. Furthermore, only the rst ve examples of those are Eu-clidean domains, whose Euclidean functions are induced by the norms; whereas,the other four have no Euclidean functions whatsoever. A brilliant proof for thelatter claim was presented by Motzkin [12] around 1949, who came up with acriterion for an integral domain to be Euclidean. But the proof seems too tersefor laymen. And lling details of Motzkin’s proof especially for non-existence ofEuclidean algorithm of ring Zh