Relatively Prime = Loopless:For {$T\in \mathcal{B}^+$}, {$T$} is relatively prime if and only there are no loops on the box with dimensions {$T$}. Proof: We prove this with a series of lemmas. The first allows us to restrict our attention to triangular boxes. {$T$} is loopless iff {$\Delta T$} is loopless: For any {$T\in\mathcal{B}^+$}, there is a loop on {$T$} iff there is a loop on {$\Delta(T)$}. Magnified Boxes Have loops: For any {$\langle a,b,c\rangle$}, {$k>1$}, the box with dimensions {$ak\times bk\times ck$} has a loop. Loopless if {$x_{_T}=0$} and {$z_{_T}=1$}: If {$x_{_T}=0$} and {$z_{_T}=1$} then there are no loops. Loops if {$x_{_T}=0$} and {$z_{_T}>1$}: If {$x_{_T}=0$} and {$z_{_T}>1$} then there are loops. Loops if {$x_{_T}>1$}: If {$x_{_T}>1$} then there are loops. We have left only the case where {$x_{_T}=1$}. If {$x_{_T}=1$}, then the final (unordered) triple after repeated applications of {$\Delta$} is {$\langle 1,y_{_T},-1\rangle$}. There are two possible {$\Delta$} pre-images of this, namely, {$\langle 1,y_{_T},y_{_T}\rangle$} and {$\langle y_{_T},y_{_T},-1\rangle$}. The case of {$\{1,y_{_T},y_{_T}\}$}: The box {$\{1,y_{_T},y_{_T}\}$} has loops iff {$y_{_T}>2$} ({$z_{_T}>1$}). The case of {$\langle y_{_T},y_{_T},-1\rangle $}: The box {$\langle y_{_T},y_{_T},-1\rangle $} has loops iff {$y_{_T}>2$} ({$z_{_T}>1$}). |