A Diophantine equation

[Kent Holing]

What is in general known about positive integer solutions (x,y) with x and y different of the equation x^4 + m x^2y^2 + y^4 = a square, for m an integer?

There are of course easy cases (such as m=+-2).

And there are known to me special cases (not easy) such that m=-1 (SIERPINSKY?) where the equation has no such solutions.

Kent Holing
NORWAY

[Randall]

Kent:

I'd suggest taking a look at Allan MacLeod's web page:
http://maths.paisley.ac.uk/allanm/ECRNT/quartic/quartint.htm

I've worked with him on this, the heights of the elliptic curves with m
negative tend to be higher than positive m for as yet unknown reason.

We found all positive solutions for 0 < m <= 20000, there are 115 as yet
unknown solutions for rank 1 curves in that range for m negative. (-20000
<= m < 0)

Randall

[Thomas Womack]

In article <pan.2006.03.03....@aol.com>,

This problem turns out to be pretty much made for the four-descent
routines which I implemented for my PhD thesis.

For example, the height-137.00987 point on the curve for m=-2075, the
first missing case, gives

x=318149603320373708357526009432116838921529941408092584231748
y=5927308819328075962581224375621489678404636415485225038855
z=53538281389389232615915082858543414319696775292938498143362410314488519691262104692722343933887502132131415873336599879

with x^4 - 2075x^2y^2 + y^4 = z^2

after 100 seconds computation on my mac mini, 62s of which was spent
using an Elkies search algorithm to find the point [783680, 277890,
1887701, -980569] on the intersection of quadrics given by the
four-descent. I'm afraid you need access to magma to use the
routines, but I'll run what cases I can here over the weekend.

If you or Allan want to get in touch with me, I'm t...@womack.net.

Tom