Number theory
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Diophantine approximation of algebraic number
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…
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Baragar-Bourgain-Gamburd-Sarnak conjecture
$latex M$ 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…