Number Systems Pertaining to Euclidean Rings of Imaginary Quadratic Integers

Abstract

Abstract. For a ring R of imaginary quadratic integers, using a concept of 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 binary or ternary number systems if and only if R is Euclidean.

Self Similar Tiles Arising From the Unitary Number Systems in Euclidean Rings of Imaginary Quadratic Integers

“…We here list some basic facts, which are necessary for our observation; the results are well known or can be found in [7].…”

“…From [7] we see that among the nine principal ideal domains of imaginary quadratic integers, only the five rings…”

“…As proposed in [7], the concept of a unitary number system can be replacement of a 'universal side divisor' by Motzkin. Furthermore unitary number systems are already available in various applications of complex based number systems.…”

“…In [7], it was shown that Euclidean rings R of imaginary quadratic integers admit unitary number systems. In this paper, as an application of the result, we obtain all self similar tiles arising from the unitary number systems of R.…”