Number theory

An important theorem in Diophantine approximation is the theorem of Liuoville: **Liuoville Theorem** If x is a algebraic number of degree $latex n$ over the rational number then there exists a constant $latex c(x) > 0$ such that:$latex \left|x-{\frac {p}{q}}\right|>{\frac {c(x)}{q^{{n}}}}$ holds for every integer $latex p,q\in N^*$ where $latex q>0$. This theorem explain a phenomenon, the approximation of algebraic number by rational number could not be very well. Which was generated la…

$latex M$ is the markov triple $latex (x,y,z)$: $latex x^2+y^2+z^2=xyz$ and $latex (x,y,x)\in \mathbb Z^3  \ \ \ \  (*)$. It is easy to see: $latex R_1: (x,y,z)\to (3yz-x,y,z)$. map markov triple to markov triple. This is also true for $latex R_2,R_3$. and the transform $latex R_1,R_2,R_3$ and permutation a classical result of markov claim that all solution of  (*) could be generated from $latex (1,1,1)$. I get a similar result for a similar algebraic equation 1 half years ago when consider a $l…