East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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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)
  • Received : 2014.09.01
  • Accepted : 2015.04.20
  • Published : 2015.05.31
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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.

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References

  1. S. Akiyama and J.M. Thuswalender, A Survey on Topological Properties of Tiles Related to Number Systems, Geom. Dedica. 109 (2004), 89-105. https://doi.org/10.1007/s10711-004-1774-7
  2. O.A. Campoli, A principal ideal domain that is not a Euclidean domain, Amer. Math. Monthly, 95 (1988), no. 9, 868-871. https://doi.org/10.2307/2322908
  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. https://doi.org/10.1109/18.119723
  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. https://doi.org/10.1090/S0002-9904-1949-09344-8
  13. W. Penney, A "binary" system for complex numbers, JACM 12 (1965) 247-248. https://doi.org/10.1145/321264.321274
  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. https://doi.org/10.1307/mmj/1028999653
  16. H.J. Song and B.S. Kang, Disclike Lattice Reptiles induced by Exact Polyominos, Fractals, 7 (1999), no. 1, 9-22. https://doi.org/10.1142/S0218348X99000037
  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. https://doi.org/10.2307/2688577