Dimensional analysis can only indicate if there might be an error but it cannot be used to provide a robust argument why something is false. Here is the problem with tegor's work. Notice that he uses r=nL. Now, greater the value of L less is the value of n, right? I will use this fact later. If you plug in K=rho/n^2 then according to his derivation you get, rho/(n^2 L). Now, although nL is equal to a constant radius 'r', greater the value of L, less is the value of n and because of that you get a higher value of R in the derived 'formula'. So, R does increase when L increases. cool?