Condensed Matter Physics (Dec 2008)
On matrices associated to prime factorization of odd integers
Abstract
In this paper we introduce in section 5 integral matrices M(n) for any factorization of an odd integer n into r distinct odd primes. The matrices appear in several versions according to a parameter ρ∈[0,1], they have size 2r×2r and their rank satisfies e.g. for ρ=1/2 the inequalities of theorem 4: r+1≤rank(M(n))≤ 2r-1+1, which are obtained using theorem 1 discussed separately in the first few sections. The cases ρ=0,1,1/2 are analyzed in some detail, and various counterexamples for ρ≠0,1,1/2 are included. There are several main results, theorem 5 is a duality between the cases ρ=0 and ρ=1, and theorem 6 is a periodicity theorem. The most important result perhaps is theorem 8 (valid for ρ=1/2 only) on the existence of odd squarefree integers n with r odd prime factors such that rank M(n)=r+1 attains the lower bound shown previously.
