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?
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> 


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](http://oeis.org/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](http://dx.doi.org/10.1090/S0025-5718-08-02168-6). ([preprint](http://www.staff.uni-bayreuth.de/~btm216/elk_ants6c.pdf))
* Apoloniusz Tyszka, *A conjecture on integer arithmetic*, Newsletter of the European Mathematical Society (75), March 2010, 56–57. ([issue](http://www.ems-ph.org/journals/newsletter/pdf/2010-03-75.pdf))
* Eric S. Rowland, *Known Families of Integer Solutions of $x^3+y^3+z^3=n$*, 2005. ([manuscript](http://www.math.tulane.edu/~erowland/papers/koyama.pdf))



Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?
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](http://arXiv.org/abs/math/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.*


