Are nontrivial integer solutions known for $x^3+y^3+z^3=3$? - MathOverflow

Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?

2011-03-11T19:05:47Z

The Diophantine equation $$x^3+y^3+z^3=3$$ has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions known at that time.

Are any other solutions known?

By a conjecture of Tyszka, it would follow that if this equation had finitely many roots, then each component of a solution tuple would be at most $2^{2^{12}/3} \lt 2^{1365.34}$ in absolute value. (To see this, it is enough to express the equation using a Diophantine system in 13 variables in the form considered by Tyszka.) This leaves a large gap, since Elsenhans and Jahnel only considered solutions with components up to $10^{14} \approx 2^{46.5}$ in absolute value. It is also not obvious whether Tyszka's conjecture is true.

OEIS sequence A173515 refers to equations of the form $x^3+y^3=z^3-n$, for $n$ a positive integer, as "Fermat near-misses". Infinite families of solutions are known for $n=\pm 1$, including one constructed by Ramanujan from generating functions (see Rowland's survey).

Andreas-Stephan Elsenhans and Jörg Jahnel, New sums of three cubes, Math. Comp. 78 (2009), 1227–1230. DOI: 10.1090/S0025-5718-08-02168-6. (preprint)
Apoloniusz Tyszka, A conjecture on integer arithmetic, Newsletter of the European Mathematical Society (75), March 2010, 56–57. (issue)
Eric S. Rowland, Known Families of Integer Solutions of $x^3+y^3+z^3=n$, 2005. (manuscript)

Answer by Luis H Gallardo for Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?

2011-03-11T19:23:22Z

Of course the problem is old and probably there is no hope to be resolved.

The following papers of Vaserstein are of interest:

MR1196532 (93k:11090) Payne, G.(1-PAS); Vaserstein, L.(1-PAS) Sums of three cubes. The arithmetic of function fields (Columbus, OH, 1991), 443–454, Ohio State Univ. Math. Res. Inst. Publ., 2, de Gruyter, Berlin, 1992. 11P05

MR1284068 (95g:11128) Conn, W.(1-PAS); Vaserstein, L. N.(1-PAS) On sums of three integral cubes. (English summary) The Rademacher legacy to mathematics (University Park, PA, 1992), 285–294, Contemp. Math., 166, Amer. Math. Soc., Providence, RI, 1994. 11Y50 (11D25) PDF Clipboard Series Chapter Make Link

Over the last 40 years there have been various computational efforts to search for integer solutions to the equation $x^3+y^3+z^3 = t$ for small integers $t$. This paper describes a search that found solutions for $t = 39$ and $t = 84$, as well as a number of other solutions

for small $t$ that are of interest for various reasons. The authors used a symbolic computation package on workstations, and used different search techniques for different regions of interest. They argue that their data supports the conjecture that solutions should exist for all $t$ satisfying the easy necessary condition that $t$ not be congruent to $\pm 4$ modulo 9; the only such $t$ less than 100 for which no solutions are known are now $30,33,42,52,74,75$. The algorithms of this paper are tuned to providing solutions for an interval of possible $t$, whereas a recent algorithm due to Heath-Brown is faster for a fixed value of $t$, although it requires significant precomputation whose complexity depends on the class number of ${\bf Q}(\root 3 \of t)$. An implementation of that algorithm by D. R. Heath-Brown, W. M. Lioen and H. J. J. te Riele [Math. Comp. 61 (1993), no. 203, 235--244; MR1202610 (94f:11132)] also discovered some of the solutions found in the article under review.

Answer by Noam D. Elkies for Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?

2011-06-02T14:31:54Z

Just noticed this question. I agree with L.H.Gallardo that the problem is old (see e.g. Problem D5 in UPINT = Unsolved Problems in Number Theory by R.K.Guy), but not that it is hopeless: the usual heuristics suggest that the number of solutions with $\max(|x|,|y|,|z|) \leq H$ should be asymptotic to a multiple of $\log H$, so further solutions should eventually emerge (though it may indeed be hopeless to prove anything close to the $\log H$ heuristic).

Answer by John R Ramsden for Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?

2011-06-02T18:09:10Z

As $x^3 + y^3 = c$ for any given (suitable) c is an elliptic curve, perhaps a reasonable strategy would be to try various integers $f$, $g$ for which $c := f^3 - 3 g^3$ is small and establish the Mordell-Weil rank of the curve.

If this is ever positive (for values other than the known solutions the OP mentioned) then one would establish that there were other non-trivial rational solutions, even if these had still not been found.

Edit: Rereading the OP's post, I notice they are asking for integer solutions rather than rational solutions, and I recall now that there are rational parametrizations anyway. So perhaps this approach isn't very useful after all.