An Algorithm For Computing Salem Number With Minimal Trace

A Salem number is an algebraic integer ?>1 of degree at least 4,conjugate to?-1,all of whose conjugates,excluding ? and ?-1.lie on the unit circle.The famous Lehmer problem is closely related to Salem number's research.Lehmer has posed the following questions:Does there exist a positive constant c such that,if M(?)>1,then M(a)>1+c?This problem can be transformed into looking for an algebraic integer with the smallest Mahler measure.The known algebraic integer with the smallest Mahler mea-sure is the largest real root of Lehmer polynomial,which is also the Salem number with the smallest trace,and Lehmer guessed that this was the smallest Salem number,so it is of great significance to find the Salem number with the smallest trace.With the properties of Salem numbers,in this paper,we construct a new auxiliary function,which improves the upper and lower bounds of the coefficients of minimal polynomials of totally real positive algebraic integers,thus reducing computing time in finding Salem numbers with minimal traces.We prove that the trace of Salem number of degree 2d=22 is greater than or equal to-1 and give all Salem numbers of degree 2d=24,26,trace(?)=-2.