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)

(as he states below.) – Tito Piezas III Dec 7 '15 at 14:52false