Thurston classified surface homeomorphisms up to isotopy. Most surface homeomorphisms are so-called pseudo-Anosov. For each pseudo-Anosov homeomorphism, there is an associated number called the stretch factor which tells us how the iterations of the homeomorphism change the length of simple closed curves on the surface (with respect to an arbitrary metric of constant curvature). We try to find a number-theoretic characterization of these numbers, and discuss the difficulty of the problem and recent partial results. This talk partially represents joint work with A. Rafiqu and C. Wu.