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On Permutations Whose Adjacent-Sum Reciprocals Sum to One
Abstract
In this paper, we give a constructive proof that for any integer n>7 (resp. n>6), the numbers 1,2,...,n can be arranged in a circle (resp. in a line) such that the sum of the reciprocals of all adjacent sums equals 1. The above conjecture was originally posed by Zhi-Wei Sun. Partitioning the integers according to n mod 12 into nine residue classes, we prove a recursive proposition and inductively construct new examples. Main lemmas and propositions in introduction are proved in Section 2. The rest of the cases are handled by explicit constructions in appendix.
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BibTeX
@misc{ai2026permutations,
title = {{On Permutations Whose Adjacent-Sum Reciprocals Sum to One}},
author = {Jinggao Ai and Dingsheng Hu and Junmin Yan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18431},
eprint = {2610.18431},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18431},
}
RIS
TY - UNPB
TI - On Permutations Whose Adjacent-Sum Reciprocals Sum to One
AU - Jinggao Ai
AU - Dingsheng Hu
AU - Junmin Yan
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18431
UR - https://arxiv.org/abs/2610.18431
AB - In this paper, we give a constructive proof that for any integer n>7 (resp. n>6), the numbers 1,2,...,n can be arranged in a circle (resp. in a line) such that the sum of the reciprocals of all adjacent sums equals 1. The above conjecture was originally posed by Zhi-Wei Sun. Partitioning the integers according to n mod 12 into nine residue classes, we prove a recursive proposition and inductively construct new examples. Main lemmas and propositions in introduction are proved in Section 2. The rest of the cases are handled by explicit constructions in appendix.
ER -