On The Linear Independence Measure Of Three Logarithms Of Rational Numbers

Let a be an irrational real number. If for any ε> 0, there exists qo(ε)> 0, such that for all integers p and q with q≥ qo(ε), then the real number μ> 0 is said to be an irrationality measure of a.Let α0, α1, …,αn be real numbers linearly independent over Q, we say that v is a linearly independence measure of α0, α1, …,αn, if for any ε> 0, there exists H0(ε)> 0, such thatIn our work, by using the integer transfinite diameter, LLL algorithm and semi-infinite linear programming, we compute the integral ∫αβf(x)/g(x)dx where the integrand f(x)/g(x) is symmetry, and f(x), g(x) G(x)∈Z[x], α,β∈Q, to research the linear independence measure of multi logarithms of rational numbers. Then we get some linear independence measures of three logarithms of rational numbers.In this paper, we also discuss the diophantine equation x2+4n= y11, and give its all the integer solutions when n is 3,4,5.