An effective p-adic analogue of a theorem of Thue II. The greatest prime factor of a binary form

Decomposable form equations

“…In view of (11) and (12) As is known (see e.g. [28], p. 71), Oa has an integral basis {wi, ... ,wdg} such that h(wi, ... ,Wdg) $ h(w1) ... h(wd 9 ) $ (lwil · ·. lwdul)dg $ c1s, (27) where lwi I denotes the maximum of the absolute values of the conjugates of Wi, i = 1, ... ,dg.…”

“…By proving and using a p-adic analogue of Baker's inequality concerning linear forms in logarithms, Coates [8], [9] made effective Mahler's theorem for irreducible binary forms F. Coates' estimate for the solutions was improved and generalized by Sprindzuk [63], [64], [66] and Shorey, van der Poorten, Tijdeman and Schinzel [57]. In (57] the authors gave an effective ,version of Mahler's theorem in full generality.…”

S-unit equations and their applications

“…Baker [l], [2] proved (implicitly) the first version of Theorem 4 (for ordinary units) and used it to make effective Thue's and Siegel's finiteness theorems mentioned above by giving explicit upper bounds for the heights of the solutions of (11.1). Coates [11], [12], in the case K = Q, and Gyory [37], [39], in the general ·case, extended these results to equation (11.l'). By using (a more explicit version of) Theorem 4, it was shown in [37], [39] that all solutions x 1 , x2 of (11.1 1 ) satisfy where c1 and c2 are positive numbers depending only on /3, F and K (which were given explicitly in [39]).…”

“…For any projective point x = (x0 : x 1 : ••• : x 12 ) in P 12 (K) and for any v E MK we put lxlv = max(lxolv, ... , lxnlv)· We define the projective height 1 ) of x as…”

A remark on the Diophantine equation (x^3-1)/(x-1)=(y^n-1)/(y-1)

“…Many authors have proved that if two of the variables x, y, m, n are fixed then the equation (2) has a finite number of solutions. See for examples [1,3,4,5,12,13,19,20,21,16,22,23,24]. Remark that two known solutions of (2) are both satisfying m = 3.…”