分野別セミナー

It is easy to check that the equation x^2+y^2+z^2=3xyz, where the three unknowns x,y,z are positive integers, has infinitely many solutions. There is a simple algorithm which produces all of them. However,this does not answer to all questions on this equation:
in particular Frobenius asked whether it is true that for each integer z>0, there is at most one pair (x,y) such that x Markoff's equation occurred initially in the study of minima of quadratic forms at the end of the XIX-th century and the beginning of the XX-th century. It was investigated by many a mathematician, including Lagrange, Hermite,Korkine, Zolotarev, Markoff, Frobenius, Hurwitz, Cassels.
The solutions are related with the Lagrange-Markoff spectrum, which consists of those quadratic numbers which are badly approximable by rational numbers. It occurs also in other parts of mathematics, in particular free groups, Fuchsian groups and hyperbolic Riemann surfaces (Ford, Lehner, Cohn, Rankin, Conway, Coxeter, Hirzebruch and Zagier...).
We discuss some aspects of this topic without trying to cover all of them.