Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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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)
  • Received : 2020.08.27
  • Accepted : 2021.10.29
  • Published : 2022.01.31
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Abstract

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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Acknowledgement

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

References

  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