A matrix representation of the primitive residue classes (mod 2𝑛)
K. Mahler · Proceedings of the American Mathematical Society · 1957
A problem1 in the geometry of numbers recently lead me to consider some simple matrices with elements 0, 1, and -1.I found to my surprise that these matrices had inverses of the same kind, that they were commutative, and that they in fact formed an Abelian group.These matrices are discussed in the present note.1. Let m and ra be two positive integers such that 1 < m < ra, (m, n) = 1.Let further i^Obea parameter and / one of its rath roots, t" = s.There are thus » distinct possible values for t, the values h, t2, ■ ■ ■ ,t" say. Now denote byA(m, ra) = (ahk) and B(m, n) = (bhk)the two raX« matrices with the following elements.For each pair of suffixes h, k = 1, 2, • ■ • , ra determine integers i, j, q, and r such that