A celebrated result in number theory is the Theorem of Markov which relates two seemingly totally different subjects: approximations of irrational numbers and the solutions of a certain equation. A proof, which Markov only sketched, was provided in detail by Frobenius precisely 100 years ago. In his paper Frobenius almost casually mentioned a problem, now known as the Uniqueness Conjecture, which has remained unsolved to this day. I will tell you about the Theorem and the Conjecture, and the amazing connections to other fields such as trees, groups, combinatorics of words, lattice paths, and matchings of plane graphs.
No knowledge in number theory or discrete mathematics beyond the basic courses is required to understand everything in the talk.