The Determination of Kaprekar Convergence and Loop Convergence of All Three-Digit Numbers

Eldridge, Klaus E.; Sagong, Seok

doi:10.2307/2323062

Cited by 5 publications

(4 citation statements)

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“…Next, Klaus E. Eldridge and Seok Sagong in their paper [11] from 1988 described the convergence of {T n 3 (x)} ∞ n=1 for all three-digit numbers x for any base r ∈ N, r ≥ 2. They obtained, among others, the following result.…”

Section: Kaprekar's Transformationsmentioning

confidence: 99%

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Kaprekar's transformations. Part I—theoretical discussion

Hetmaniok

Pleszczyński

Sobstyl

et al. 2015

Annals of Computer Science and Information Systems

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Abstract-The paper is devoted to discussion of the minimal cycles of the so called Kaprekar's transformations and some of its generalizations. The considered transformations are the self-maps of the sets of natural numbers possessing n digits in their decimal expansions. In the paper there are introduced several new characteristics of such maps, among others, the ones connected with the Sharkovsky's theorem and with the Erdős-Szekeres theorem concerning the monotonic subsequences. Because of the size the study is divided into two parts. Part I includes the considerations of strictly theoretical nature resulting from the definition of Kaprekar's transformations. We find here all the minimal orbits of Kaprekar's transformations Tn, for n = 3, ..., 7. Moreover, we define many different generalizations of the Kaprekar's transformations and we discuss their minimal orbits for the selected cases. In Part II (ibidem), which is a continuation of the current paper, the theoretical discussion will be supported by the numerical observations. For example, we notice there that each fixed point, familiar to us, of any Kaprekar's transformation generates an infinite sequence of fixed points of the other Kaprekar's transformations. The observed facts concern also several generalizations of the Kaprekar's transformations defined in Part I.

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Section: Kaprekar's Transformationsmentioning

confidence: 99%

“…Formulae (11) and (12) describe the following 45 numbers 111 × 9, 121 × 9, 222 × 9, 232 × 9, 242 × 9, 333 × 9, 343 × 9, 353 × 9, 363 × 9, 444 × 9, 454 × 9, . .…”

mentioning

confidence: 99%

Kaprekar's transformations. Part I—theoretical discussion

Hetmaniok

Pleszczyński

Sobstyl

et al. 2015

Annals of Computer Science and Information Systems

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“…In septenary, since 6 2 ×3=108=213 (7) , so calculating 48 positive integers within 108, in which a digit in high place is not greater than that in a low place, we know the iteration has five fixed points: 1, 10=13 (7) , 25=34 (7) ,32=44 (7) ,45＝63 (7) , and two 4-circles (2,4, 16, 8)=(2, 4, 22, 11) (7) , (13, 37, 29, 17)=(16, 52, 41, 23) (7) . Proportions for converging to seven cases are about 1/48, 11/48, 3/16, 1/48, 1/8, 1/4, 1/6 respectively.…”

Section: Iteration Of Sum Of Squares Of Digitsmentioning

confidence: 99%

“…In septenary, since 6 3 ×4=864=2343 (7) , so computing 158 positive integers which their a digit in higher position is less or equal than a lower one within 864, we know the iteration has seven fixed points: 1, 9=12 (7) , 16=22 (7) , 133=250 (7) , 134=251 (7) , 152 ＝ 305 (7) , 250 ＝ 505 (7) , four 2-circles (2,8)=(2, 11) (7) , (91, 217)=(160, 430) (7) , (92, 218)=(2, 11) (7) , (244, 496)=(466, 1306) (7) , two 3-circles (10, 28, 64)=(13, 40, 121) (7) , (36, 126, 72)=(51, 240, 132) (7) , one 4-circles (258, 342, 648, 282)= (516, 666, 1614, 552) (7) , and one 9-circles ( 17, 35, 125, 251, 341, 557, 137, 197, 65)=(23, 50, 236, 506, 665, 1424, 254, 401, 122) (7) In octal, since 7 3 ×4=1372=2534 (8) , so computing 249 positive integers which their a digit in higher position is less or equal than a lower one within 1372, we know the iteration has six fixed points: 1, 92=134 (8) , 133=205 (8) , 307=463 (8) , 432=660 (8) , 433=661 (8) , and one 5-circle (434, 440, 559, 469, 476)=(662, 670, 1057, 725, 734) (8) . Proportions for converging to seven cases are about 59/249, 9/83, 4/249, 1/83, 5/83, 5/249, 136/249 respectively.…”

Section: Iteration Of Sum Of Cubes Of Digitsmentioning

confidence: 99%

Periodic Points Under Iteration of Sum of Squares or Cubes of Digits in Some Positional Systems

Wu¹,

Dang²,

Zhang³

2015

Proceedings of the 3rd International Conference on Material, Mechanical and Manufacturing Engineering

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Abstract. There are different periodic and fixed points under iteration of sum of squares or cubes of digits of positive integer in different positional systems. On the iteration of sum of squares of digits, in binary and quaternary these points all converge to fixed point 1. There are three fixed points and one 2-circle in ternary, three fixed points and one 3-circle in quinary, one fixed points and one 8-scircle in senary, five fixed points and two 4-circles in septenary, three fixed points and two 2-circles and one 3-circle in octonary, three fixed points and one 2-circle and one 3-circle in nonary, one fixed point and one 6-circle in hexadecimal. On the iteration of sum of cubes of digits, in binary these points all converge to fixed point 1. There are two fixed points and one 4-circle in ternary . There are only nine fixed points in quaternary. There are three points and one 3-circle in quinary, four fixed points and one 5-circle in senary, seven fixed points, four 2-circles, two 3-circles, one 4-circles and one 9-circlein septenary, six fixed points and one 5-circle in octonary, eight fixed points, two 2-circle, one 5-circle and one 11-circle in nonary, 24 fixed point, two 2-circles, one 3-circle, one 6-circle, two 10-circles, one 14-circle and one 15-circle in hexadecimal. The longest circle is found in base-14 and it is a 27-circle.

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Searching for Kaprekar′s constants: algorithms and results

Walden

2005

International Journal of Mathematics and Mathematical Sciences

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The Determination of Kaprekar Convergence and Loop Convergence of All Three-Digit Numbers

Cited by 5 publications

References 2 publications

Kaprekar's transformations. Part I—theoretical discussion

Kaprekar's transformations. Part I—theoretical discussion

Periodic Points Under Iteration of Sum of Squares or Cubes of Digits in Some Positional Systems

Searching for Kaprekar′s constants: algorithms and results

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