Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

LIPTAI, K. and TÓMÁCS, T., Pure powers in recurrence sequences

36 Kálmán Liptai and Tibor Tómács Another type of problems was studied by Ribenboim and McDaniel, For a sequence R we say that the terms R

m

, R

n are in the same square-class if there exist non zero integers x, y such that Rm% — RnV 1 or equivalently RmRji — t 5 where t is a positive rational integer. A square-class is called trivial if it contains only one element. Ribenboim [16] proved that in the Fibonacci sequence the square-class of a Fibonacci number F

m is trivial, if m ^ 1,2, 3, 6 or 12 and for the Lucas sequence L( 1,1,2,1) the square-class of a Lucas number L

m is trivial if m / 0,1,3 or 6. For more general sequences R(A

, B

, 0,1), with (A, B

) = 1, Ribenboim and McDaniel [17] obtained that each square class is finite and its elements can be effectively computed (see also Ribenboim [18]). Further on we shall study more general recursive sequences. Let G = G(Ai

,..., A/-, G

0, • • •, G/e-i

) be a k

t

h order linear recursive sequence of rational integers defined by Gn = A\G

n—i + A

2G

n2 + • • • + A

kG

nk (n > k - 1), where and Go, • • •, Gk-\ are not all zero integers. Denote by a = cti

, a

2,..., a

s the distinct zeros of the polynomial x

k — A\X

k~

l — A

2x

k~

2 — • • • — Afc. Assume that a, a

2,..., a

s has multiplicity l,m

2,..., m

s respectively and |a| > |ct^

] for i = 2,..., s. In this case, as it is known, the terms of the sequence can be written in the form (1) G

n = aa

n + r

2(n)a% + • • • + r

s(n)< > 0), where T{(i = 2, ...,ő) are polynomials of degree m; — 1 and the coefficients of the polynomials and a are elements of the algebraic number field Q(a, c*2,

..., a

s). Shorey and Stewart [21] prowed that the sequence G does not contain q

t

h powers if q is large enough. This result follows also from [7] and [22], where more general theorems where showed. Kiss [6] generalized the square-class notion of Ribenboim and McDaniel. For a sequence G we say that the terms G

m and G^n. are in the same q

t

hpower class if G

mG

n ~ w

q

, where w,q rational integers and q > 2. In the above mentioned paper Kiss proved that for any term G

n of the sequence G there is no terms G

m such that m > n and G

n, G

m are elements of the same g

t

h-power class if q sufficiently large.