To my Wife Wahida

In 1997, Andrew Beal [1] announced the following conjecture: Let A,B,C,m, n, and l be positive integers with m,n, l> 2. If Am +Bn = Cl then A,B, and C have a common factor. We begin to construct the polynomial P (x) = (x − Am)(x − Bn)(x + Cl) = x3 − px + q with p, q integers depending of Am, Bn and Cl. We resolve x3 − px+ q = 0 and we obtain the three roots x1, x2, x3 as functions of p, q and a parameter θ. Since Am, Bn,−Cl are the only roots of x3 − px + q = 0, we discuss the conditions that x1, x2, x3 are integers. A numerical example is given