We prove that every contact form on the tight three-sphere has at least two geometrically distinct periodic orbits. This result was obtained recently by Cristofaro-Gardiner and Hutchings using embedded contact homology but our approach instead is based on cylindrical contact homology. An essential ingredient in the proof is the notion of a symplectically degenerate maximum for Reeb flows whose existence implies infinitely many prime periodic orbits (in any dimension). This is joint work with V. Ginzburg, D. Hein and U. Hryniewicz.