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 $n\geq10$, there exist infinitely many $n\times n$ integer matrices $A$ such that every entry is $\ge2$ 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 $n\geq10$.