A086446 extension?

[James Buddenhagen]

Thanks Hugo for posting Dave Rusin's interesting comments. It 
appears to me that Allan MacLeod has been pretty thorough in 
computing the data 
( http://maths.paisley.ac.uk/allanm/ecrnt/knight/knposres )
upon which A085514 is based and, since this sequence is not 
easy to compute perhaps this sequence should be extended in 
OEIS up to 999, or maybe better provide a link directly to 
MacLeod's data (the link above) and not just to the parent page. 
It is still not clear to me whether all values of N (between 11 
and 999) missing from MacLead's list are proved not to belong or 
whether perhaps some are absent based on assumed truth of standard 
elliptic curve conjectures. 

With regard to the derived subsequence A086446 of David Wilson's 
inquiry, this part of Rusin's July 30, 2003 email to Pfoertner is 
relevant:

> It looks again like the positivity condition on  a,b,c  is equivalent to
> there being a rational point in the "egg" (i.e. having  X < 0)  and again
> I believe the points not on the egg form a subgroup of index 2, so the
> n's with positive solutions can be spotted immediately when we have
> generators of  E / 2E . Well, here are the generators (X,Y) I found, and 
> the ones with X<0 are the ones on your list A086446. I will take the fact
> that these lists match as a proof that this is the right characterization
> of positivity :-)

 From the last quoted sentence I conclude that A086446 is conjecturally 
correct but not proven correct.  The paper of Bremmer et al might shed more 
light on this, but I haven't seen that paper either.  Rather than extend 
A086446 in OEIS based on MacLeod's data, I would favor the more cautious 
approach of stating that it is conjectured that A086446 excludes all N in 
MacLead's data for which y is negative, and then give the link to MacLeod's 
data.

Jim Buddenhagen