From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Simultaneous Pellian Equations
Date: 17 Aug 1998 00:00:00 GMT
Newsgroups: sci.math

Radford de Peiza <rdepeiza@interlog.com> wrote:
>I am working on a problem which requires me to solve a pair of
>simultaneous 'Pellian' equations. I have found relatively little
>information on this topic. Can someone provide references? The two
>equations are 3x^2 -y^2 = 2 and 2x^2 - z^2 = 1 where x, y and z are
... let's be generous here and let them be rational numbers.

A quadratic equation in three variables determines a conic surface in
3-space. Two such equations determine the intersection of such surfaces;
except in special degenerate cases, these intersections are elliptic
curves (once a "basepoint" is singled out). Then we need to know if
this elliptic curve has any rational points apart from the obvious
(e.g. a point at x=y=z=1, which I'll use as a basepoint).

The answer is "no". We may parameterize the first curve (write
x=1+x' and y=1+t*x', then solve for x') so that all rational values
for x are obtained as a ratio of quadratic polynomials in  t. Then
the second equation simply asks that  2x^2-1  be a square; its denominator
is, so we only need the numerator -- a quartic polynomial in  t  -- to
be a square.

From here it's easiest to reach for software: the Maple package  APECS
can take the quartic-with-rational-point, recognize it as an elliptic
curve, compute its torsion group (of order 8) and attempt -- in this case
successfully -- to compute its rank, here zero. The layman's interpretation
is that only for t=0, 1, or 3 (or t="infinity") is the quartic a rational
square. These values lead only to x= +- 1 (and y=+1, z=+-1).

Had there been infinitely many rational points, it would have been a little
trickier deciding what all the integral points were, but on general
principles there could only have been a finite number of them.

For the two-conics-make-an-elliptic-curve ideas, see Cassell's book on
elliptic curves. For the only-finitely-many-integral-points material, you'll
need e.g. Silverman's book. Some other pointers and so on are at
	index/14H52.html

dave