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On the existence of Graham partitions with congruence conditions
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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.
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.
참고문헌 (4)
- M. A. Alekseyev, "The mathematics of various entertaining subjects. Vol. 3", pp.213~221 (2019), Princeton Univ. Press
- G. E. Andrews, "The theory of partitions, reprint of the 1976 original, Cambridge Mathematical Library" (1998), Cambridge University Press
- R. L. Graham, "A theorem on partitions", J. Austral. Math. Soc, 3, pp.435~441 (1963)
- B. Kim, "On the partitions into squares whose reciprocal sum is one", Publ. Math. Debrecen, 95(1-2), pp.243~247 (2019)
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