Number Theory Seminar

Speaker: Laurent Vuillon (Univ Savoie Mont Blanc).

Title: Combinatorics on words for Markoff numbers.

Abstract: Markoff numbers are fascinating integers related to number theory, Diophantine equation, hyperbolic geometry, continued fractions and Christoffel words. Many great mathematicians have worked on these numbers and the 100 years famous uniqueness conjecture by Frobenius is still unsolved. In this talk, we state a new formula to compute the Markoff numbers using iterated palindromic closure and the Thue-Morse substitution. The main theorem shows that for each Markoff number m, there exists a word $v\in\{a, b\}^*$ such that $m − 2$ is equal to the length of the iterated palindromic closure of the iterated antipalindromic closure of the word $av$. This construction gives a new recursive construction of the Markoff numbers by the lengths of the words involved in the palindromic closure.