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
6 P. Kiss & F. Mátyás for any i (1 < i < m). It is known that there exist uniquely determined polynomials p^\x) G Q(a ( 1' ', a^\ ..., (j — 1,2,..., t{) of degree less than (i) (O the multiplicity rrv- of the roots a- such that for n > 0 (3) (?£> - ( a (py +1 4°(n) (*<<>)" + . . . + (*'•>)" . In that special case when g^ l\x) has a dominant root, say CE; = A^ , that LS, when the multiplicity of a; is 1 and |a,;| > a^ ' for j = 2, 3,. . . ,ti, then [a ?;| > 1, > 1, and Pi\n) in (3) is a constant which will be denoted by a;. In since this case 4? (4) &•<» = a, («,)" + ("':') " + •••+ p\%) Ur where we suppose that ai ^ 0. We say to be the dominant sequence among the sequences G^ (1 < i < hold for any m) if (x) has a dominant root a\ and the inequalities |q i| > (i,j) ^ (1,1), where 1 < i < m and 1 < j < ti. (0 0 1 j T. N. Shorey and C. L. Stewart [13] investigated the connection between the sequences defined by (4) and perfect powers, then A. Pethő [11], [12] and P. Kiss [6] proved important results in this field. Recently, some similar multiplicative and additive problems have been solved by B. Brindza, K. Liptai and L. Szalay [3], L. Szalay [14], P. Kiss and F. Mátyás [8-9] and F. Mátyás [10]. All of the authors show, under some restrictions, that if a term (product or sum of terms) of linear recurrences is a perfect power then the exponent of the power is bounded above. The problem is similar when we want to consider those sequences where the terms of G^ have given prime factors only. Let pi,po, • • • ,p r be given distinct rational primes and let (5) S = {se Z : s = ±pl 1 ... p\7, 0 < e t- G N} . K. Győry, P. Kiss and A. Schinzel [4] showed that if G x Is a term of Lucas or Lehmer (special second order) recurrences then (6) G x G 5' holds only for finitely many sequences and finitely many integers x. K. Győry [5] improved this result.
