Someone recently asked me how to proof that $x+1$ and $x^3$ generate a free group. A colleague has worked out a proof. I have a vague memory that this has been studied, maybe a Monthly problem? Does anyone know any history on this?
Edit: Sorry for my omission of key point. The group operation here is function composition on the reals, or the integers. Each of the polynomials can be viewed as a permutation of Z = all integers (or on the reals). Viewed in that way, do they generate a free group (of rank two).