From: rusin@shuksan.math.niu.edu (Dave Rusin)
Subject: Re: 2909  =  | n^2 - m^3 | / 2^k   ?
Date: 23 Jan 1999 07:56:42 GMT
Newsgroups: sci.math
Keywords: Integral points on more elliptic curves

John Langley <jlangley@quanset.com> wrote:
>Is 2909 the odd part of the difference between a square and a cube?

Probably not, if you mean for  n, m  to be integral. But
If  x = 363641/3025  and  y = 225083811/166375  then (y^2-x^3)/32 = 2909.
Solutions are given below for some other powers of 2, and it can be
shown that for other powers of two, there is no rational solution,
let alone an integral one.


The equation to be solved may be written
	y^2 = x^3 +- 2^k*2909
For any fixed  k  and choice of sign, this describes an elliptic curve.
Two such curves with the same sign are birationally equivalent iff the
corresponding  k  are congruent mod  6, so up to equivalence, there
are  12  curves to consider. All of them are torsion-free.


		Case 1: y^2 = x^3 + 2^k*2909:

For k=0 we have a curve whose rank over the rationals I could not determine.
Connell's program  APECS computes Mestre's upper bound on the rank (with
parameter 5000) to be 2.07 and we expect an even rank, so either there are no 
points or there's a rank-2 group of them. I guess I'd bet there are none.
The question may be asked, are there rational points on the curve
y^2 = 5 x^4 + 4 x^3 + 12 x^2 + 27 x + 3. I had Cremona's RATPOINT look 
for solutions with a limit parameter of  8; nothing found.

k=1: The curve has rank  1. I didn't check to see if this is a generator or
not but I'd guess it is; at least it _is_ on the curve:
	Point = [-2524894515 : 27079544783 : 365525875]
That is, if P = [a,b,c], then x = a/c, y=b/c. These and other completed
rank calculations performed with MWRANK.

k=2, 3, 4: rank is zero (no rational points)

k=5: rank is 1, an (assumed) generator is
	Point = [20000255 : 225083811 : 166375]


		Case 2: y^2 = x^3 - 2^k*2909:

k=0: rank is 1, an (assumed) generator is
	Point = [1426167990 : 4708460867 : 75151448]

k=1: rank=1 conjecturally, but no generator found. Need a solution to
	y^2 = 5x^4 + 8x^3 - 114x^2 + 64x - 191
and one must exist barring an unforeseen failure of important conjectures,
but none found even setting RATPOINT parameter to  10 !

k=2,3,4,5: rank=0.