From: pethoe@neumann.math.klte.hu (Dr Petho Attila) Subject: Re: Squares in Integer Linear Recurrences Date: 5 Oct 00 16:38:02 GMT Newsgroups: sci.math.numberthy Summary: [missing] For systematic study of perfect powers in second order recurrences I recommend the original papers: {\sc A. Peth�,} Perfect Powers in Second Order Linear Recurrences, J. Number Theory, {\bf 15} (1982), 5-13. {\sc M. Mignotte} and {\sc A. Peth\H{o},} {\em On the system of diophantine equations $x^2-6y^2=-5$ and $x=2z^2-1$,} Math. Scandinavica, {\bf 76} (1995), 50--60. {\sc T.N. Shorey} and {\sc C.L. Stewart,} {\em On the Diophantine equation $ax^{2t} + bx^ty + cy^2 = d$ and pure powers in recurrences,} Math. Scand., {\bf 52} (1983), 24--36. and the book {\sc T.N. Shorey} and {\sc R. Tijdeman,} {\em Exponential Diophantine Equations}, Cambridge Univ. Press 1986. About perfect powers in higher order recurrences much less is known. See {\sc P. Corvaja} and {\sc U. Zannier}, {\em Diophantine equations with power sums and universal Hilbert sets}, Indag Math., N.S. {\bf 9} (1998), 317-332. {\sc P. Corvaja} and {\sc U. Zannier}, {\em Some new applications of the subspace theorem}, manuscript. and {\sc A. Peth\H{o},} {\em Diophantine properties of linear recursive sequences. II.}, Acta. Math. Acad. Paed. Ny\'iregyh\'aziensis, submitted. (You can find here a proof that if G_n is a trird order linear recurrence such that its characteristic polynomial is irreducible and has a dominating root, then there are only finitely many perfect powers in it. This result is not effective.) Attila Pethoe > From owner-nmbrthry@LISTSERV.NODAK.EDU Thu Oct 5 17:10:39 2000 > X-External-Networks: yes > Approved-By: "Victor S. Miller" > Date: Thu, 5 Oct 2000 12:06:57 -0400 > From: "Victor S. Miller" > Subject: Squares in Integer Linear Recurrences > To: NMBRTHRY@LISTSERV.NODAK.EDU > > I have the following problem: > > Define the sequence > > u[n] = F[n+1]^2 + 4 * F[n] > > where n is the n-th Fibonacci number. It appears (and I've checked > this by sieving for n < 2^22), that the only value of n for which u[n] > is a square is n = 0. First, u[n] satisfies a 5-th degree linear > recurrence. Second, I know of the result of JHE Cohn who proved that > except for a few small values that (I think 1 and 144) no member of > F[n] is a square, but this proof is rather ad hoc. I would think that > there is a way to reduce this problem to a problem in linear forms in > logarithms. I came across a paper by Shorey and Stewart about perfect > q-th powers in integer linear recurrences, but they specifically > exclude squares. So can anyone help out with some ideas about how to > prove this? > -- > Victor S. Miller | " ... Meanwhile, those of us who can compute can hardly > victor@idaccr.org | be expected to keep writing papers saying 'I can do the > CCR, Princeton, NJ | following useless calculation in 2 seconds', and indeed > 08540 USA | what editor would publish them?" -- Oliver Atkin >