Hall's conjecture concerns large integral points on elliptic curves
of the special form *y*^{2}=*x*^{3}+*b*.
What about large integral points on the general curve
*y*^{2}=*x*^{3}+*ax*+*b*?
(This is not quite the most general elliptic curve over **Z**,
but putting an arbitrary curve in this form introduces
only bounded factors into all the estimates.) It turns out that,
whereas for *y*^{2}=*x*^{3}+*b*
one expects that *x*<<*b*^{2+o(1)},
there are curves
*y*^{2}=*x*^{3}+*ax*+*b*
with *a* constant that have an integral point with
*x* asymptotically proportional to *b*^{4}.

Standard heuristics suggest that if we consider curves with
a<<*H*^{2} and b<<*H*^{3}
then *x* can be as large as *H*^{10+o(1)} but no larger.
Zagier addressed this question in [3]; the addendum [3a]
reports that Lang and H.Stark already conjectured in [2]
that *x*<<*H*^{R}
for all *R*>10,
but with the proviso that some families might have *x*
growing as a power of *H* higher than the tenth.
Soon after reading Zagier's paper I found an example of such a family
with *x* asymptotically proportional to *H*^{12},
and as it happens with a constant linear coefficient *a*.
To my knowledge, this family and its easy variations are still
the only such families known.

I wrote Zagier a letter giving this construction,
which he acknowledged in the addendum [3a].
It hasn't been published otherwise, though the remarks
concerning Hall's conjecture were subsequently incorporated
into my paper [1] (see p.56). The letter can now be viewed
in PDF
or .dvi;
please note the sign error on page 2:
the second curve in the family has
(*t*, *b*)=(-15, 17424),
not (15, -17424) as written.

**References**

[1] Elkies, N.D.: Rational points near curves and small nonzero
|*x*^{3}-*y*^{2}| via lattice reduction,
* Lecture Notes in Computer Science* **1838**
(proceedings of ANTS-4, 2000; W.Bosma, ed.), 33-63
(math.NT/0005139 on the arXiv).

[2] Lang, S.: *Arithmetic and Geometry*, Vol.I, 155--171
[Progr. Math. **35**, Birkhäuser, Boston, Mass., 1983].

[3] Zagier, D.: ``Large integral points on elliptic curves'',
*Math. Comp.* **48** (1987) #177, 425-436.

[3a] Zagier, D.: Addendum to [3],
*Math. Comp.* **51** (1988) #183, page 375.