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Bull. Korean Math. Soc. 2022; 59(1): 15-25

Published online January 31, 2022 https://doi.org/10.4134/BKMS.b200730

Copyright © The Korean Mathematical Society.

On the existence of Graham partitions with congruence conditions

Byungchan Kim, Ji Young Kim, Chong Gyu Lee, Sang June Lee, Poo-Sung Park

Seoul National University of Science and Technology; Seoul National University; Soongsil University; Kyung Hee University; Kyungnam University

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.

Keywords: Graham partition, sum of reciprocals, Rogers--Ramanujan identity

MSC numbers: Primary 11P81, 05A17

Supported by: Byungchan Kim and Poo-Sung Park were supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (NRF-2019R1F1A1043415, NRF-2017R1A2B1010761). Ji Young Kim, Chong Gyu Lee and Sang June Lee were supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2020R1I1A1A01053318, NRF-2016R1D1A1B01009208, NRF-2019R1F1A1058860).

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