Condensed Matter Physics, 2008, vol. 11, No. 4(56), p. 723747
DOI:10.5488/CMP.11.4.723
Title:
On matrices associated to prime factorization of odd integers
Author(s):

T. Bier
( DOMAS, Faculty of Science, Sultan Qaboos University, Muscat 123, AlKhod PO Box, Oman ; Twiskenweg 43 B, D 26129 Oldenburg, Germany)

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 2^{r}×2^{r} and their rank satisfies e.g. for ρ=1/2 the inequalities of theorem 4: r+1≤rank(M(n))≤ 2^{r1}+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.
Key words:
factorization, matrices, floor, ceiling, square roots of unity
PACS:
02.10.Yn
