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

Pure powers in recurrence sequences KÁLMÁN LIPTAI* and TIBOR TÓMÁCS Abstract. Let G be a linear recursive sequence of order k satisfying the recursion G„ = A

1G

n_H M

f

cG„_jfc. In the case k=

2 it is known that there are. only finitely many perfect powers in such a sequence. Ribenboim and McDaniel proved for sequences with /c=2, G

0= 0 and G^—l that in general for a term G

n there are only finitely many terms G

m such that G

nG

m is a perfect square. P. Kiss proved that for any n there exists a number q

0, depending on G and n, such that the equation G

nG

x=w

q in positive integers x,w,q has no solution with x>n and q>qo

• We show that for any n there are only finitely many x\,x

2,--.,Xk ,x,w,q positive integers such that G

n G

X

l •••G

X

f, G

x =w

q and some conditions hold. Let R = R(A, B

, Rq, Ri) be a second order linear recursive sequence defined by R

n = AR

n-i + BR

n-2 (n > 1), where A, B, Rq and Ri are fixed rational integers. In the sequel we assume that the sequence is not a degenerate one, i.e. aj

ß is not a root of unity, where a and ß denote the roots of the polynomial x

2 — Ax — B. The special cases R( 1,1, 0,1) and R(

2,1, 0,1) of the sequence R is called Fibonacci and Pell sequence, respectively. Many results are known about relationship of the sequences R and perfect powers. For the Fibonacci sequence Cohn [2] and Wylie [23] showed that a Fibonacci number F

n is a square only when n = 0,1, 2 or 12. Pethő [12], furthermore London and Finkelstein [9,10] proved that F

n is full cube only if n = 0,1, 2 or 6. From a result of Ljunggren [8] it follows that a Pell number is a square only if n — 0,1 or 7 and Pethő [12] showed that these are the only perfect powers in the Pell sequence. Similar, but more general results was showed by McDaniel and Ribenboim [11], Robbins [19,20] Cohn [3,4,5] and Pethő [15]. Shorey and Stewart [21] showed, that any non degenerate binary recurrence sequence contains only finitely many perfect powers which can be efHctively determined. This results follows also from a result of Pethő [14]. Research supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA T 16975 and 020295.