From: kesslert@surfree.com (Troy Kessler)
Subject: A Problem of Selmer
Date: 3 May 01 19:22:38 GMT
Newsgroups: sci.math.numberthy
Summary: Elliptic curves X^3+Y^3=m*Z^3 with many rational points

In a paper of Selmer, he did calculations on the curves X^3+Y^3=m*Z^3 and
calculated the rank for m<500. He did this by considering curves of the
form a*x^3+b*y^3+c*z^3=0 where abc=m (a,b,c)=1 and a<b<c. In another paper
he states that m=1020 has 13 out of 13 soluble abc equations. I have found
a example of 40 out of 40 soluble abc equations. The example is m=131070
and the solutions of the abc equations are give below.
a b c x y z
1 2 65535 1 -32 1
1 3 43690 -157 71 4
1 5 26214 -53 -23 2
1 6 21845 -1321 611 35
1 10 13107 413 -200 9
1 15 8738 13 -9 1
1 17 7710 -11 -23 3
1 30 4369 -719 133 41
1 34 3855 -79 -29 7
1 51 2570 181 -61 13
1 85 1542 -23 5 1
1 102 1285 143 -101 43
1 170 771 -1019 -152 129
1 255 514 -329 9 41
1 257 510 -199 -7 25
2 3 21845 -164 -189 11
2 5 13107 386 -443 29
2 15 4369 -23 11 1
2 17 3855 59 -29 1
2 51 1285 -11 3 1
2 85 771 -7 -1 1
2 255 257 -1 -1 1
3 5 8738 179 -151 1
3 10 4369 -251 -83 23
3 17 2570 -19 1 2
3 34 1285 2511 -1118 23
3 85 514 -28547 9947 -2999
3 170 257 -49 4 11
5 6 4369 1 -9 1
5 17 1542 -1517 1001 64
5 34 771 229 -122 13
5 51 514 374123 33541 -80059
5 102 257 1279 -183 -337
6 17 1285 -22 -131 31
6 85 257 12 -5 1
10 17 771 2609 -6931 1923
10 51 257 -62 23 19
15 17 514 15 -17 4
15 34 257 1 -2 1
17 30 257 -1 -2 1
Does anyone know a fast algorithm to search for solutions on curves of abc
type? I think 1588620 has 121 out of 121 but have not finished the
calculation. If one could figure out a pattern and generate m's with
(3^g-1)/2 out of (3^g-1)/2 soluble abc curves one could create a curve of
the form X^3+Y^3=m*Z^3 with rank g. If a pattern exists this would prove
that elliptic curves of arbitrary rank exist.
==============================================================================

   16,14g 10.0X
   Selmer, Ernst S.
   A conjecture concerning rational points on cubic curves. 
   Math. Scand. 2, (1954). 49--54.

   The author reports that using an electronic computor he has found
   solutions for all the equations for which he could give no decision in
   an earlier memoir [Acta Math. 85, 203--362 (1951); these Rev. 13, 13]
   except in two cases where he can prove insolubility by different
   criteria. This and other numerical evidence to be presented elsewhere
   later leads to interesting conjectures. If the L.H.S. of the
   Diophantine equation $x\sp 3+y\sp 3=az\sp 3$ is factorised in the
   Eisenstein field of cube roots of 1 and coefficients are equated,
   there result a finite number of equations of the type $w\sp 3=f(u,v)$
   where $f(u,v)$ is a ternary cubic with rational integer coefficients.
   The number of these which are soluble is closely related to the number
   of generators of the group of rational solutions in the usual sense.
   By congruence considerations equations $w\sp 3=f(u,v)$ may often be
   shown to be insoluble. One can now factorise $f(u,v)$ in the relevant
   cubic fields and so obtain new equations of the type
   $\phi(u',v',w')=0$, where $\phi$ is a ternary cubic with integer
   coefficients. These can be again sometimes eliminated by congruence
   considerations ("second descent"). The author remarks that in a large
   number of cases an even number of generators is eliminated by the
   second descent and conjectures that this always happens. Indeed he
   conjectures that when a second descent exists the difference between
   the number of generators allowed by the first descent and that
   actually existing is even. He also shows that a second descent can be
   made with $Y\sp 2=X\sp 3-AX-B$ and makes similar conjectures.

                        Reviewed by J. W. S. Cassels

   (c) 2001, American Mathematical Society