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. Pethơ,} 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" <victor@IDACCR.ORG>
> Date:         Thu, 5 Oct 2000 12:06:57 -0400
> From: "Victor S. Miller" <victor@idaccr.org>
> 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
>