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Reducing Every Set of 36 Consecutive Integers to Zero by Differences of Squares
Abstract
For a finite multiset of integers, repeatedly choose two entries a,b and replace them with |a2−b2|. Hickerson and Kleber asked whether, for every integer n, the set n,n+1,…,n+35 can be reduced to zero. We give an explicit reduction. Together with their results for lengths 12 and 24, this gives, for every positive integer L, [ {n,n+1,\ldots,n+L-1}\text{ reduces to zero for every }n\in\mathbb Z \Longleftrightarrow 12\mid L\text{ and }L\ge 24. ]
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BibTeX
@misc{shen2026reducing,
title = {{Reducing Every Set of 36 Consecutive Integers to Zero by Differences of Squares}},
author = {Zhao Shen and Youran Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00098},
eprint = {2609.00098},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00098},
}
RIS
TY - UNPB
TI - Reducing Every Set of 36 Consecutive Integers to Zero by Differences of Squares
AU - Zhao Shen
AU - Youran Wu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00098
UR - https://arxiv.org/abs/2609.00098
AB - For a finite multiset of integers, repeatedly choose two entries a,b and replace them with |a^2−b^2|. Hickerson and Kleber asked whether, for every integer n, the set n,n+1,…,n+35 can be reduced to zero. We give an explicit reduction. Together with their results for lengths 12 and 24, this gives, for every positive integer L, [ {n,n+1,\ldots,n+L-1}\text{ reduces to zero for every }n\in\mathbb Z \Longleftrightarrow 12\mid L\text{ and }L\ge 24. ]
ER -