On matrices associated to prime factorization of odd integers - Bier, T. (Condensed Matter Physics, 2008)
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 ϵ 2 [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.