ON A CODİNG METHOD OF LOCKED NUMBERS by rfm80781

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									                  ON A CODİNG METHOD OF LOCKED NUMBERS
                       Urfat Nuriyev urfat.nuriyev@ege.edu.tr
                                     Ege Üniversitesi, Department of Mathematics

         The notion of “locked number” was firstly defined by Dj.A.Babayev [1] in
2004. This notion is generalized of the number 6174 (Kaprekar’s constant) for
other digit lengths, which took Indian mathematician, Kaprekar 3 years to calculate
in 1946 [2,3].Other researchers examined this kind of numbers for different digit
lengths [4-6]. Results of ongoing research on Kaprekar cycles are presented in [7].
         The study [8] is about the use of locked numbers in information
technologies (cryptology, computer games,etc.) , whereas in study [9] , some
characteristics of locked numbers are investigated, a computer game program
based on these characteristics is designed and all locked numbers up to 70-digits
are calculated.
         In this study, a coding diagram is proposed due to respresentation of
locked numbers in study [10] and a table is prepared to shortly present locked
numbers within 70- digits.When examining the table, it is observed that the
following structures repeat in different forms :
 a1=5, a2=(5,0) ; b1 =(6,2), b2=(8,6,4,2); c1=(9,7,5,1), c2=(7,5,1); d=(7,7,5,4,3,2,0).
         Using these notations, we can code locked numbers up to 70- digits and
obtain a new table.It is seen from table that locked numbers have the following 6
structures:

      a32k  a1k  a2 k                                                              (1)
      a32k 1  a1k  a2 k 1                                                       (2)
      b2k  b1  3 k 2 , k  2                                                        (3)
      b   b  3  0 , m  1, k  9
        2
            2m
                          2
                            2m        k 9 m      m
                                                                                         (4)
      b   b  3  0 , m  0, k  4
               2k
          2 m1                   2 m1       k 9 m4     m
        2         2 k 1        2                                                        (5)
      b d   b  d  0 , m  1, k  1
          m
                9 m14k
                                    m       k       m
                                                                                        (6)
      b c c   b  c  c  9  3  0 , p  q  k  9l  3m  4, l  0, m  0, k  4
        2                         2
          21        m                 21             m     p       q       l
        2     1 2                   2        1     2                                     (7)
      b c c   b c c  9  3  0 , p  q  k  9l  3m  8, l  0, m  0, k  8
                         2k
          21l         m                   21l        m       p       q       l
        2        1 2       2 k 1        2        1 2                                    (8)

       These structures provide determining locked numbers analitically and they
make characteristics of them search easily.

References:
1 Dj A. Babayev , Locked Numbers, September, 2004, http://www.cox-
associates.com/djangir/LockedNumbers.doc
2. D. R. Kaprekar, “Another Solitaire Game”, Scripta Math. 15 (1949) 244-245.
3. D. R. Kaprekar, “An Interesting property of the number 6174”, Scripta Math. 21
(1955), 304.
4. J. H. Jordan, Self-Producing Sequences of Digits, Amer. Math. Monthly 71
(1964) 61-64.
5. J. F. Lapenta, A.L. Ludington, G. D. Prichett, Algorithm to Determine Self
Producing r- Digit g-Adic Integers. J. Reine Angew. Math. 310 (1979) 100-110
6. G. D. Prichett, A.L. Ludington, J.F. Lapenta, The Determination of All Decadic
Kaprekar Constants” Fibonacci Quarterly 19.1 (1981) 45-52
7. Kaprekar Series Generator, http://kaprekar.sourceforge.net
8. Babayev Dj.A., Nuriyev U.G., Özarslan S, Sözeri V. On locked numbers, XIX.
National Mathematics Symposium, August 22-25, 2006, Kütahya .
9. Babayev Dj.A., Nuriyev U.G., Özarslan S., Locked numbers and their
application in information technologies , International scientific conference
“Information Technologies and Telecommunications in Education and Science”
(IT & T ES’2005), p.190-194, Antalya, Turkey, May 15-22, 2005.
10. Babayev Dj.A., Nuriyev U.G., Amrenova D.A., On a representation of locked
numbers, International scientific conference “Information Technologies and
Telecommunications in Education and Science” (IT & T ES’2007), Fethiye,
Turkey, May 18-25, 2005.

								
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