>From owner-nmbrthry@LISTSERV.NODAK.EDU Tue Jul 9 16:54:56 1996
Apparently-To:
From: Noam Elkies
Subject: x^3 + y^3 + z^3 = d
To: Multiple recipients of list NMBRTHRY
Representing an integer d as a sum x^3+y^3+z^3 of three integer cubes
is a long-standing problem. It is known that this cannot be done for
d congruent to 4,5 mod 9 (clearly) or d=0 (Euler); there are infinitely
many polynomial solutions whenever d is a cube, and finitely many when
d is twice a cube, e.g. the only known polynomial solution for d=2 is
(x,y,z)=(1+6t^3+1,1-6t^3,-6t^2) [with permutations of x,y,z or changes
of variable in t consider to be the same]. There are no analytic
results for any other d, though the usual heuristics suggest that
given d (not 4 or 5 mod 9) solutions should occur infinitely often,
but rarely, with asymptotically c*log(N) solutions in |x|,|y|,|z|1)
whose only known representations came from representations of d' by
scaling, and several d for which only one representation was known,
e.g. 12 = 7^3 + 10^3 - 11^3. R.Guy wrote with more recent references:
W.Conn and L.N.Vaserstein (in "The Rademacher Legacy to Mathematics",
Contemp. Math. 166 (1994)) and D.R.Heath-Brown, W.M.Lioen and H.J.J.
te Riele (Math.Comput.1993) found several new solutions, including
2 = 1214928^3 + 3480205^3 - 3528875^3
(the first one not accounted for by the above polynomial) and
39 = 117367^3 + 134476^3 - 159380^3
(39 was the third-smallest unknown value of d, the first two being
30 and 33), and more recently (mid-1995) Richard F. Lukes found
several new solutions such as
110 = 109938919^3 + 16540290030^3 - 16540291649^3
with x,y,z rather large but y+z small. The methods were:
1) Extending the exhaustive search for small values of x^3+y^3=z^3,
which takes time N^2 (ignoring logarithms) and logarithmic space;
2) For specific d, solve for each potential value of y+z the congruence
x^3 == d mod y+z and search over x satisfying this congruence. This
takes time N, and can also efficiently find larger solutions such as
Lukes' above with y+z small, but only gets solutions for one d at a
time and in practice requires more space (to sieve over y+z) and,
at least as implement by Heath-Brown et al., auxiliary conditions
on the arithmetic of Q(cbrt(d)).
Some weeks ago I realized that it is possible to find all solutions of
|x^3+y^3+z^3|<<|x|+|y|+|z|<