Studies On The Absolute Length And The Absolute Mahler Measure Of Totally Positive Algebraic Integer

For totally positive algebraic integerαof degree d,C.J.Smyth[1,2]and V. Flammang[3,4]considered the set E of values of L(α)^1/d = R(α) and the set L of values of M(α)^1/d=Ω(α),where L(α) is the length ofαand M(α) is the Mahler measure ofα.V.Flammang proved that there are only six elements of E in the interval I₁ =[2,2.3611014) and six elements of L in the interval I₂ =(1,1.720566). In this work,we improve V.Flammang's theorem,that is to say,we improve the right endpoint of I₁ up to 2.364556 and the right endpoint of I₂ up to 1.721899.Finally,we discuss the Diophantine equation x³ - 1 = 103y² and give its all the integer solutions[5].