TITLE: Minimal length presentations of cyclic groups
QUESTION [12 upvotes]: By the length of a finite presentation I mean the sum of the lengths of the relators.  I am interested in knowing what the minimal length of a presentation of $\mathbb{Z}/n\mathbb{Z}$.  I'm even more interested in knowing the minimal length of a balanced presentation $\mathbb{Z}/n\mathbb{Z}$.  
For concreteness, I am really interested in the particular case of $\mathbb{Z}/173\mathbb{Z}$.  For example this has a presentation of length 24 given by $<a,b,c | a^4c, a^3b^5, a^{-2}bc^8>$ - is this a minimal length presentation? Is this presentation minimal length among balanced presentations?

REPLY [7 votes]: I can beat that by one.
$$C_{173} \cong \langle a,b,c,d,e \mid a^4=b, b^2=c,c^4=d,d^4=e, abcde=1 \rangle.$$
It might be hard to prove minimality. I am sure the best you can do is $O(\log n)$, but it could be interesting to see what constant you could get. The  approach of using squares of successive generators together with the binary expansion of $n$ gives a bound of $4 \log_2 n$, which could be reduced to about $3.5 (\log_2 n+1)$ by using positives and negatives in the binary expansion of $n$.
But I think using cubes and the ternary expansion of $n$ is better in general. That seems to give a bound of $5 (\log_3 n +1)\sim 3.155 \log_2 n+5$.