Greatest common denominator of measurements In a couple months, I'll do the Millikan experiment. Then, I'll end up with a number of charge measurements and their errors $$((q_i, \Delta q_i))_{i \in \mathbb N}.$$ The idea is that all those $q_i$ can be represented as a multiple of a fixed elementary charge $e$ like $$q_i = n_i e.$$
Now I do not know $e$ in advance. Is there some method to get $(e, \Delta e)$ out of the sequence of measurements?
I found this question on StackOverflow, but it does not have any answers that solve the problem in a stable way.
 A: This is how you would do it if there were no experimental error:  Let $q_0$ be your smallest measured value for the charge and enumerate the other charges $q_1,\dotsc,q_n$.  By assumption, $q_k=en_k$ for some positive real $e$ and $n_k$ an integer.  Thus, $\frac{q_k}{q_0}$ is rational for all $k$.  Write all of these rationals is reduced form and let $\ell$ be the least common multiple of all the denominators.  Take $e=q_0/\ell$.  Then, by construction, $q_k/e$ is an integer for all $k$ and $e$ is the smallest positive real number such that this is true.
The problem with this, however, is that, when you calculate $\frac{q_k}{q_0}$, you are only getting an estimate, and so $\ell$ is going to be completely off.  In practice, I think you just have to hope that $\frac{q_k}{q_0}$ is (approximately) an integer for all $k$, and if that's not the case, go back to the lab and take more data until this is the case.
Of course, as pointed out, if $e$ works, so will $e/2$, and $e/3$, etc.  The best you will ever be able to do here is calculate the probability that you measured your value of $e$ (call it $e_0$) was indeed the correct value of $e$ under the assumption the true value of $e$ is $e_0/2$, $e_0/3$, etc.
A: Generally, computing approximate integer gcds is hard. In fact, a couple of years ago an encryption scheme was introduced whose security depends on such hardness, see Fully Homomorphic Encryption over the Integers by van Dijk, Gentry et al. Chasing references to this should turn up much of interest.
