The Smallest Perron Numbers

A Perron number ? is a real algebraic integer with degree d ? 2,whose conjugates are ?i,such that ?>(?)|?_i|.For the smallest Perron number,there is a famous conjecture of Lind-Boyd[9]:Conjecture(Lind-Boyd).The smallest Perron number of degree d? 2 has minimal polynomialThis conjecture was studied by many people for many years.All the smallest Perron number of degree d<25 satisfy this conjecture.In our work,we improve the algorithm for computing the upper and lower bounds of S_k,we then have the smallest Perron number of degree d = 26,27,and verified this Perron number satisfies the conjecture of Lind-Boyd.