TITLE: Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?
QUESTION [33 upvotes]: 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)

REPLY [9 votes]: 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.