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

Question

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 1

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).

See also my article

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.

Among other things it gives an algorithm for finding all solutions of $|x^3 + y^3 + z^3| \ll H$ with $\max(|x|,|y|,|z|) \leq H$ that should run (and in practice does run) in time $\widetilde{O}(H)$; since we expect the number of solutions to be asymptotically proportional to $H$, this means we find the solutions in little more time than it takes to write them down.

D.J.Bernstein has implemented the algorithm efficiently, and reports on the results of his and others' extensive computations at http://cr.yp.to/threecubes.html .

EDIT: for the specific problem $x^3+y^3+z^3=3$, Cassels showed that any solution must satisfy $x\equiv y\equiv z \bmod 9$ in this brief article:

A Note on the Diophantine Equation $x^3+y^3+z^3=3$, Math. of Computation 44 #169 (Jan.1985), 265-266.

This uses cubic reciprocity, and is stronger than what one can obtain from congruence conditions. See also Heath-Brown's paper "The Density of Zeros of Forms for which Weak Approximation Fails" (Math. of Computation 59 #200 (Oct.1992), 613-623), where he gives corresponding conditions for the homogeneous equation $x^3 + y^3 + z^3 = 3w^3$ and also $x^3 + y^3 + z^3 = 2w^3$, and reports that

In a letter to the author, Professor Colliot-Thélène has shown that the above congruence restrictions are exactly those implied by the Brauer-Manin obstruction. Moreover, for the general equation $x^3 + y^3 + z^3 = kw^3$, with a noncube integer $k$, there is always a nontrivial obstruction, eliminating two-thirds of the adèlic points.

Answer 2

My conjecture is false, see

Apoloniusz Tyszka, All functions $g:\mathbb{N}\to\mathbb{N}$ which have a single-fold Diophantine representation are dominated by a limit-computable function $f\colon \mathbb{N}\setminus \{0\}\to\mathbb{N}$ which is implemented in MuPAD and whose computability is an open problem, Computation, cryptography, and network security (eds. N. J. Daras and M. Th. Rassias), Springer, 2015, pp. 577–590, doi:10.1007/978-3-319-18275-9_24, arXiv:1309.2682

Answer 3

The solution given in the comment of Y. Zhao:

$569936821221962380720^3 + (-569936821113563493509)^3 + (-472715493453327032)^3 = 3$.

Note that each of these numbers is larger than the limit of $10^{14}$ at which Elsenhans and Jahnel stopped their search in 2007. The smallest has 18 digits, while the other two have 21.

Answer 4

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 5

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.