[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.7858/eamj.2015.027

NUMBER SYSTEMS PERTAINING TO EUCLIDEAN RINGS OF IMAGINARY QUADRATIC INTEGERS

Sim, Hyo-Seob (Department of Applied Mathematics, Pukyong National University)
Song, Hyun-Jong (Department of Applied Mathematics, Pukyong National University)

Publication Information

East Asian mathematical journal / v.31, no.3, 2015 , pp. 357-362 More about this Journal

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.

Keywords

Euclidean rings; imaginary quadratic integers; Complex base number systems; Fractals;

Citations & Related Records

Reference

1	S. Akiyama and J.M. Thuswalender, A Survey on Topological Properties of Tiles Related to Number Systems, Geom. Dedica. 109 (2004), 89-105. DOI
2	O.A. Campoli, A principal ideal domain that is not a Euclidean domain, Amer. Math. Monthly, 95 (1988), no. 9, 868-871. DOI ScienceOn
3	K. Conrad, Remarks about Euclidean domains, an expository paper in Ring Thery.
4	K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, Wiley, (1990).
5	D. Goffinet, Number systems with a complex base: a fractal tool for teaching topology, Amer. Math. Monthly 98 (1991), no. 3, 249255.
6	L. Guillen, A principal ideal domain that is not a Euclidean domain, a personal note in website.
7	K. Groechenig and W.R. Madych, Multiresolution Analysis, Haar bases and self-similar tilings of $R^n$ , IEEE Trans. Inform. Th. 38(2) Part2 (2)(1992), 556-568. DOI ScienceOn
8	S.I. Khmelnik, Specialized digital computer for operations with complex numbers, Questions of Radio Electronics (in Russian) XII (2)(1964).
9	S.I. Khmelnik, Positional coding of complex numbers, Questions of Radio Electronics (in Russian) XII (9)(1966).
10	D.E. Knuth, An Imaginary Number System, Communication of the ACM-3 (4) (1960).
11	N. Koblitz, CM-curves with good cryptographic properties, Advances in Cryptology-CRYPTO '91', LNCS 576, 1992, 279-287.
12	T. Motzkin, The Euclidean Algorithm, Bull. Amer. Math. Soc., 55 (1949), 1142-1146. DOI
13	W. Penney, A "binary" system for complex numbers, JACM 12 (1965) 247-248. DOI
14	A. Petho, On a polynomial transformation and its application to the construction of a public key cryptosystem, Computational Number Theory, Proc., Walter de Gruyter Publ. Comp., Eds.: A. Petho and etals, (1991), 31-44.
15	H.M. Stark, A complete determination of the complex quadratic fields of class number one, Michigan Math. J. 14 (1967), 1-27. DOI
16	H.J. Song and B.S. Kang, Disclike Lattice Reptiles induced by Exact Polyominos, Fractals, 7 (1999), no. 1, 9-22. DOI ScienceOn
17	I. Stewart, D. Tall, Algebraic Number Theory, Chapman and Hall Mathematics Series, Second Edition.
18	K.S. Williams, Note on non-Euclidean principal ideal domains, Amer. Math. Monthly , 48(1975), no. 3, 176-177.
19	Jack C. Wilson, A Principal Ring that is Not a Euclidean Ring, Math. Mag. 46 (Jan 1973), 34-38. DOI