[Update 1999/06/22: I see I mistyped the problem in the analysis! The poster
 asked about  d = 41*73  and I provided an answer but stated I was
 analyzing the case  d = 43*71. I was, in fact, analyzing the correct
 problem. However, for completeness I now also address the case  d=43*71
 by the same analysis, at the end of the file. -- djr]
==============================================================================

From: rusin@olympus.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Diophanto Eq
Date: 31 Dec 1996 19:26:49 GMT

sfly (b841039@math.ntu.edu.tw) wrote:
> how to solve the eq.......x^4-d*y^2=1 where d=41*73
> and , is ther there rational numbers  a,b,c such that
> a^2-b^2=b^2-c^2=d    (d=41*73)

There are no nontrivial solutions in either case, although this
conclusion is harder to come by than is typical in these questions.

Hauke Reddmann <fc3a501@GEO.math.uni-hamburg.de> wrote:

>Elliptic Curves (move x=1,y=0 to infinity)

>a+b=d/l
>a-b=l
>b+c=d/m
>b-c=m
>d/l-l=d/m+m
>Which is again an elliptic curve after a bit
>variable rewriting.

This is the way to go.

Let's look at the second system of equations first. As Hauke noted,
the equations may be rewritten  	d/l-l=d/m+m ;
upon multiplying by  l*m  we obtain a cubic equation which may be
expressed as an elliptic curve in standard form. Indeed, using the
invertible transformation {l=d*(X-d)/Y, m=X*(X-d)/Y} the equation becomes

                            2
                           Y  = X (X - d) (X + d)

This equation clearly has solutions with Y=0 (namely   X=0,
X= +- d, and the "point at infinity"). But these don't correspond to 
solutions  (l,m)  of Hauke's pair of equations, nor to solutions
(a,b,c) of the original pair.

Now, the points on an elliptic curve form a finitely-generated abelian
group; these four points are merely the torsion subgroup. What's really 
interesting is the question of whether this group has positive rank.

There are a number of algorithms which enable us to estimate the rank
of a general elliptic curve over the rational field, but these don't
give sufficiently strong upper bounds for our particular curve.
However, curves of the form described above are precisely those
involved in Tunnell's resolution (sort of) of the "congruent number
problem". Tunnell proves that there is a more-or-less complete
algorithm which can tell us at least whether the rank of the curve is
positive or not.

Here's the test (for  d  square-free). Count the number of integer solutions 
(x,y,z) to the equation
	2x^2 + y^2 + 8z^2 = d.
Let  c1  be the number of such solutions. Let  c2  be the number of
such solutions with  z  even. Then if the elliptic curve  y^2=x(x^2-d^2)
has positive rank, we must have c1 = 2 c2. (The converse is probably true,too).

Unless I miscounted, there are for  d=43*71  exactly  c1=96  solutions but 
only  c2=40  with  z  even, so by Tunnell's theorem, the rank of this elliptic 
curve is zero. The only points are the torsion points, and so there are no
solutions to the original equations.

Now we can address sfly's other equation. Again, Hauke's suggestion is fine:
using the invertible transformation {y=4*Y/(X-2*d)^2, x=(X+2*d)/(X-2*d)}
the equation becomes
                                2       2        2
                               Y  = X (X  + (2 d) )

While this curve is not of the type covered directly by Tunnell's theorem,
it happens to be isogenous to the first curve we considered. (That is,
there are homomorphisms  E -> E'  and  E' -> E  whose composites are the
doubling maps on each curve). In particular, these two elliptic curves
have the same rank, which we have shown to be zero. Thus again the only
points on this curve are the torsion points, which can be checked to be
only the point (X,Y)=(0,0)  and the point at infinity (unless d=1). 
In terms of the original coordinates, this means the only rational solutions 
are (x,y) = (+-1, 0).

Of course, if you start with a different value of  d, then the situation
can be quite different. I'm glad you didn't suggest  d=157.

dave

==============================================================================

[Update 1999/06/22: Here we treat an alternative problem. -- djr]

If  d = 43*71, then c1=c2=0 so we expect these curves to have positive rank.
A simple computation with APECS or MWRANK finds the
elliptic curve in question to have rank 1, and finds an element of
infinite order (presumably a generator of the torsion-free part of the
Mordell-Weil group):
	P = [169771267/14161, 2139128217120/1685159].

Working backwards from this one solution leads to  l=10248637/476880, 
m=2410231/28560, a=11258184984239/137672394720, b = 8299464974161/137672394720,
and c = -3318961972561/137672394720.  There are infinitely many solutions,
of course, found by carrying out the substitutions starting from all points
P' = n.P + T  for any integer  n  and any of the four torsion points  T.

Since this curve has rank 1, it follows that the other curve 2-isogenous
to it also has rank 1; applying a 2-isogeny to the known point  P  provides
a point on this curve too:  
  Q = [626817319142400/55910021209, 17869773239282241366720/13220092244931677]
which we substitute back to find a solution 
     x = 11258184984239/3318961972561  
     y = 2285214435775015585659840/11015508575306004116898721
to  x^4-d*y^2=1 where d=43*71.