From: gwalsh@MATHSTAT.UOTTAWA.CA (Gary Walsh)
Subject: Re: Diophantine question
Date: 21 Nov 00 20:46:06 GMT
Newsgroups: sci.math.numberthy
Summary: [missing]

On Wed, 8 Nov 2000 16:49:55 -0500, Thomas W Cusick
<cusick@acsu.buffalo.edu> wrote:

>Consider the recursion y_n = 4 y_{n-1} - y_{n-2} for n = 0, 1, ...
>with y_0 = 4 and y_1 = 9.  Is there any reasonably simple way to prove
>that two consecutive members of the sequence y_0, y_1, ... are not
>both squares, other than the initial pair 4, 9?
>
>Tom Cusick

It is not hard to see that a square in this sequence is equivalent
to a solution to the quartic equation X^2-3Y^4=-47, and also, two
consecutive elements in the sequence being squares is equivalent
to a positive integer solution to the Thue equation
X^4-4X^2Y^2+Y^4=-47 (the consecutive squares 4 and 9 above correspond to
the solution (X,Y)=(3,2)). Therefore, any "reasonable" method to solve
this question evidently solves this Thue equation. There are ways to deal
with such problems, but perhaps not as reasonable as the querent desires.

For the record, solving equations such as X^2-3Y^4=-47 is described
in great detail in Tzanakis' paper [Acta Arith. 75 no. 2 (1996) 165-190].

My question is: for d squarefree and k fourth-power free, is there
an absolute bound for the number of integer solutions to quartic
equations of the form X^2-dY^4=k? More simply: Is there a heuristic
argument which indicates that the number of integer solutions to such
equations is unbounded?

Gary Walsh