Simultaneous Unimodularity of Positive Integer Matrices and Their Entrywise Cubes
Abstract
We solve an open problem posed by Richard Guy in his book Unsolved Problems in Number Theory. We prove that, for every integer n≥10, there exist infinitely many n× n integer matrices A such that every entry is ≥2 and det A=det A[3]=1, where A[3] denotes the entrywise cube of A. The matrices may be chosen compatibly, so they can form the leading principal blocks of a single infinite positive integer array. The order-ten case is reduced to an inhomogeneous system consisting of one linear and one diagonal cubic equation. The circle method then provides infinitely many positive integral solutions. To pass from order n to order n+1, we introduce a bordering construction that preserves the signed cofactor vectors of both the matrix and its entrywise cube. Iterating this construction yields the result in every order n≥10.
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BibTeX
@misc{li2026simultaneous,
title = {{Simultaneous Unimodularity of Positive Integer Matrices and Their Entrywise Cubes}},
author = {Xinyi Li and Shuochen Wang and Zimeng Ye and Alexandru Zaharescu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14546},
eprint = {2610.14546},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14546},
}
RIS
TY - UNPB
TI - Simultaneous Unimodularity of Positive Integer Matrices and Their Entrywise Cubes
AU - Xinyi Li
AU - Shuochen Wang
AU - Zimeng Ye
AU - Alexandru Zaharescu
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14546
UR - https://arxiv.org/abs/2610.14546
AB - We solve an open problem posed by Richard Guy in his book Unsolved Problems in Number Theory. We prove that, for every integer n≥10, there exist infinitely many n× n integer matrices A such that every entry is ≥2 and det A=det A^[3]=1, where A^[3] denotes the entrywise cube of A. The matrices may be chosen compatibly, so they can form the leading principal blocks of a single infinite positive integer array. The order-ten case is reduced to an inhomogeneous system consisting of one linear and one diagonal cubic equation. The circle method then provides infinitely many positive integral solutions. To pass from order n to order n+1, we introduce a bordering construction that preserves the signed cofactor vectors of both the matrix and its entrywise cube. Iterating this construction yields the result in every order n≥10.
ER -