Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)
Kiss, P. & MÁTYÁS, F., On products and sums of the terms of linear recurrences
10 P. Kiss & F. Mátyás if ,Ti > nu • According to (5) and the notation y — max (ej), l<i<T e c" r i > |s| = JJp? 1' > 2 y = i = 1 that is, (23) Let A log a 1a 1' for A, as follows: y = max (eA < cisx'i. l<t<r . By (11a), A / 0. From (22) we can obtain an upper estimation (24) 0 < A < log (1 + e~ ClAX l) < e~ ClAX l. To construct a lower estimation for A we apply Lemma 1 for A = T : e t- log pi - log |«i| - xi log laj I We can similarly get, as in the proof of Theorem 1, that (25) A > e _Cl8 log:r i . Making a comparison between (24) and (25), we get Cl3®l < etc log Xi, from which xi (26) log a?] < Ci7 follows. This proves the theorem, since (26) is a contradiction if > n 1 2, that is, the theorem holds with 0 < c\ < cio and ni = max (nj). 7<i<12 The statements of the corollaries are obvious by the theorems. References [1] BAKER, A., A sharpening of the bounds for linear forms in logarithms II., Acta Arithm., 24 (1973), 33-34. [2] BAKER, A., WUSTHOLTZ, G., Logarithmic forms and group varietes, J. Reine Angew. Math ., 442 (1993), 19-62.
