Pure powers in recurrence sequences
Kálmán Liptai, Tibor Tómács · Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) · 1997
Let G be a linear recursive sequence of order k satisfying the recursion G" = A 1 G n _H M fc G"_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 n G 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 n G 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 Xl •••G Xf , G x =w q and some conditions hold.