I want to show that there can't be simple loops in the Collatz conjecture over positive numbers, where you began with an odd number n, then after applying $\frac{3n+1}{2}$ then you repeat the process until you get to an even number. When it arrives at an even number the number is in the form $\mathrm{2}^cn$ (c integer). So dividing by $\mathrm{2}^c$ will complete the loop. (Execpt case $n=1$)
What I have tried: Write n in the form $n=\mathrm{2}^b(2a-1)-1$ (n can be written in this form since any odd number can be written in this form.)
Applying $\frac{3n+1}{2}$ yeilds: $\frac{3((2a-1)\mathrm{2}^b-1)+1}{2} = \frac{3(2a-1)\mathrm{2}^b-3+1}{2} = \frac{3(2a-1)\mathrm{2}^b-2}{2} = {3(2a-1)\mathrm{2}^{b-1}-1}$
Which is still odd if $b>1$ and this process can be applied b times before an even number is output. Which yields $\mathrm{3}^b(2a+1)-1$
The desired form of the loop gives: $\mathrm{3}^b(2a+1)-1=\mathrm{2}^c(\mathrm{2}^b(2a+1)-1)$ I believe showing there are no solutions in natural numbers (besides the case corresponding to n=1) would solve the problem, but I am unable to find a way to prove that.
Can it be shown that there is no solutions to this equation or is there an easier approach?
Ray P Steiner, A theorem on the Syracuse problem, Proc. 7th Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba-Winnipeg 1977), Congressus Numerantium XX, Utilitas Math. (1978) 553-559, MR 80g:10003 proved that the only positive loop of the type under discussion is the one starting at $1$ (allowing negative integers, there's also the one starting at $-1$, and the one starting at $-5$). According to Lagarias' annotated bibliography, the problem reduces to finding positive integer solutions to $$ (2^{k+\ell}-3^k)h=2^{\ell}-1. $$ Steiner shows that the only solution in positive integers is $(k,\ell,h)=(1,1,1)$. The proof uses results from transcendence theory, Baker's method of linear forms in logarithms (which won Baker a Fields medal, so you see what you are up against).
Many have tried to find Steiner's paper on the web, no one I know of has succeeded. Interlibrary loan should work. Also, no one I know of has succeeded in finding a proof simpler than Steiner's.