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On the minimal elements for the sequence of all powers in the Lemoine-Kátai algorithm


Author: Jukka Pihko
Journal: Math. Comp. 60 (1993), 425-430
MSC: Primary 11B83; Secondary 11Y55
DOI: https://doi.org/10.1090/S0025-5718-1993-1155575-9
MathSciNet review: 1155575
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Abstract: It is proved, with the help of a computer, that for $ m = 20$ the first m minimal elements for the sequence of all powers in an integer-representing algorithm are given by $ {y_i} = i,i = 1,2,3,{y_{i + 1}} = (y_i^2 + 6{y_i} + 1)/4,i = 3, \ldots ,m - 1$. This extends an earlier result of the author (for $ m = 10$).


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  • [1] A. S. Fraenkel, Systems of numeration, Amer. Math. Monthly 92 (1985), 105-114. MR 777556 (86d:11016)
  • [2] I. Kátai, Some algorithms for the representation of natural numbers, Acta Sci. Math. (Szeged) 30 (1969), 99-105. MR 0240039 (39:1393)
  • [3] -, On an algorithm for additive representation of integers by prime numbers, Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 12 (1969), 23-27. MR 0262196 (41:6806)
  • [4] -, On additive representation of integers, Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 13 (1970), 77-81. MR 0319925 (47:8466)
  • [5] E. Lemoine, Décomposition d'un nombre entier N en ses puissances $ n$ièmes maxima, C.R. Paris XCV (1882), 719-722.
  • [6] -, Sur la décomposition d'un nombre en ses carrés maxima, Assoc. Franç. Tunis 25 (1896), 73-77.
  • [7] -, Note sur deus nouvèles décompositions des nombres entiers, Assoc. Franç. Paris 29 (1900), 72-74.
  • [8] G. Lord, Minimal elements in an integer representing algorithm, Amer. Math. Monthly 83 (1976), 193-195. MR 0398961 (53:2812)
  • [9] M. B. Nathanson, An algorithm for partitions, Proc. Amer. Math. Soc. 52 (1975), 121-124. MR 0379353 (52:258)
  • [10] J. Pihko, An algorithm for additive representation of positive integers, Ann. Acad. Sci. Fenn. Ser. A. I Math., Dissertationes No. 46 (1983), 1-54. MR 700564 (84k:10042)
  • [11] -, On a question of Tverberg, Théorie des Nombres (Quebec, PQ, 1987), de Gruyter, Berlin-New York, 1989, pp. 806-810. MR 1024605 (90i:11019)
  • [12] -, Fibonacci numbers and an algorithm of Lemoine and Kátai, Applications of Fibonacci Numbers (G. E. Bergum et al., eds.), Kluwer Academic Publishers, Dordrecht, 1990, pp. 287-297. MR 1125801 (92j:11016)
  • [13] -, On sequences having same minimal elements in the Lemoine-Kátai algorithm, Fibonacci Quart. 30 (1992), 344-348. MR 1188738 (94a:11028)
  • [14] P. Ribenboim, Consecutive powers, Exposition. Math. 2 (1984), 193-221. MR 783135 (86h:11025)
  • [15] E. S. Selmer, Private communication, May 21, 1991.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1993-1155575-9
Article copyright: © Copyright 1993 American Mathematical Society

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