Fw: a/b + b/c + c/a = n

[all at abouthugo.de]

Weitergeleitete Nachricht von rusin at math.niu.edu vom 15.07.2003,
03:02:33:
> 
> I just took a closer look at your message. You seem to
> be implying that having multiple solutions for a particular  n
> is unusual. That is not true. For n>5, when there is a solution,
> there are infinitely many. For example, when  n=6  the solutions
> include  {t,u,v} = {1,2,3}, {1817, 3258, 5275}, {4904676969, 10840875082,
> 15051171563}, etc. These can be computed easily by taking odd multiples
> of the generator on the elliptic curve. (I restrict to odd multiples
> only because you have asked for t,u,v  to be positive; it's not obvious
> but not too hard to check that this is the necessary condition.)
> For another example, here is the next triple for   n=9:
> {t,u,v} = 970703,2982043,4461282
> 
> The case n=5 is exceptional because in that case the additional 
> solutions come from larger-than-generic torsion in the curve; the
> free rank is zero, so the only solution is {t,u,v}={1,1,2}.
> 
> The case n=3 is also exceptional because the curve is of genus zero,
> which means we can actually parameterize the set of all solutions.
> It turns out they are of the form  x/z = (-1/2) + (m/9), y/z= (-1/2) - (m/9)
> where  m  can be any rational number; choose  z  to be any multiple of the
> denominators of these two expressions.
> 
> > numbers with multiple solutions are 6, 69,77,94,105,129,149,154,161(3),174,...
> 
> I am not at all surprised to report that in each of these cases,
> the rank of the corresponding elliptic curve is (well, appears to be) 2.
> (The computation of the rank of an elliptic curve frequently involves
> making appeal to some very deep conjectures. These conjectures are
> still unproven, but their failure would be headline news in algebraic
> geometry, so I feel quite confident asserting that their rank is  2.)
> I did not check the intermediate cases to determine whether any of
> them had rank 2 as well. But it looks like you are finding these
> second examples (t,u,v) simply because the extra generator can be
> expressed with numbers of much smaller height than the triple of the
> other generator, that is, these were low fruits ready for the picking!
> I suppose, for example, that for  n=69  you found these two solutions
> which correspond to the two generators of low height: {2,57,73} and
> {42,95,523}. But other combinations corresponding to (n1)(gen.1)+(n2)(gen.2)
> also work, as long as the multipliers (n1)+(n2) have an odd sum (to
> ensure positivity of  t,u,v). For example, 2(gen.1)+(gen.2) turns out
> to correspond to the solution {t,u,v}={38808119,45866266,349822755}.
> 
> I suppose I should add another technical caveat: when I refer to 
> "generators", I am describing a finitely-generated abelian group (with
> known and very small torsion). It is technically possible that the
> things I call "generators" are merely generating a proper subgroup,
> of finite index. (Think about finding the two elements g1=(1,1) and 
> g2=(1,-1) in the group Z^2.) It would require a slow, exhaustive
> search to rule out the very unlikely event that these generators I
> have found are merely generating such a subgroup. In particular, it
> is possible that in some of the cases where I noted "the rank is 1 but
> the generator does not yield positive (t,u,v)", that an astounding
> coincidence has taken place, and this "generator" is actually twice (say)
> a generator, and that true generator _does_ lead to positive (t,u,v).
> I maintain that this would be extremely unlikely, but I did not
> bother to rule it out entirely.
> 
> Elliptic curve theory is much better suited to finding extraordinary
> examples than to disproving the existence of such things...
> 
> dave
> 
> PS - you are again welcome to do anything with this message that you like.