대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제59권1호
  • /
  • Pages.15-25
  • /
  • 2022
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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ON THE EXISTENCE OF GRAHAM PARTITIONS WITH CONGRUENCE CONDITIONS

  • Kim, Byungchan (School of Liberal Arts Seoul National University of Science and Technology) ;
  • Kim, Ji Young (Department of Mathematical Sciences Seoul National University) ;
  • Lee, Chong Gyu (Department of Mathematics Soongsil University) ;
  • Lee, Sang June (Department of Mathematics Kyung Hee University) ;
  • Park, Poo-Sung (Department of Mathematics Education Kyungnam University)
  • 투고 : 2020.08.27
  • 심사 : 2021.10.29
  • 발행 : 2022.01.31
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초록

In 1963, Graham introduced a problem to find integer partitions such that the reciprocal sum of their parts is 1. Inspired by Graham's work and classical partition identities, we show that there is an integer partition of a sufficiently large integer n such that the reciprocal sum of the parts is 1, while the parts satisfy certain congruence conditions.

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과제정보

The authors are indebted to the referee for comments and corrections, which improved the presentation of this paper.

참고문헌

  1. M. A. Alekseyev, On partitions into squares of distinct integers whose reciprocals sum to 1, in The mathematics of various entertaining subjects. Vol. 3, 213-221, Princeton Univ. Press, Princeton, NJ, 2019.
  2. G. E. Andrews, The theory of partitions, reprint of the 1976 original, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998.
  3. R. L. Graham, A theorem on partitions, J. Austral. Math. Soc. 3 (1963), 435-441. https://doi.org/10.1017/S1446788700039045
  4. B. Kim, J. Y. Kim, C. G. Lee, and P. Park, On the partitions into squares whose reciprocal sum is one, Publ. Math. Debrecen 95 (2019), no. 1-2, 243-247. https://doi.org/10.5486/pmd.2019.8574