What is the solution of $x^3+y^3+z^3=429$ in integers? I have tried to solve  $x^3+y^3+z^3=429$  using mathematica (Reduce[x^3+y^3+z^3 == 429 && x > 0 && y > 0,&& z > 0,  {x, y,z}, Integers] ) and wolfram alpha I can't come up to the solution of $x^3+y^3+z^3=429$ however $429$ mod $9$ neither $5$ or $4$ mod $9$  and in the same time is not classified as unsolved as montioned in linked papers  which are montioned in  this MO-question, , Probably I have missed somethings in my Code , ,I was interested to this number to know more about Catalan numbers representation as sum of three cubic.
 A: Known integer solutions as of Apirl 2007${}^{\color{blue}{[1]}}$.
$$\begin{align}
429 &= 284^3 + 182^3 +(-307)^3\\
    &= 644^3 + 533^3 + (-748)^3\\
    &= 871146950^3 + 15204917^3 + (-871148494)^3
\end{align}$$
Notes/References


*

*$\color{blue}{[1]}$ - 
List of solutions of $x^3 + y^3 + z^3 = n$ for $n < 1000$
neither a cube nor twice a cube. 
Andreas-Stephan Elsenhans and Joerg Jahnel, April 2007 (
an online copy of the list can be found here).

A: May be this will help (This way you can search a range). I don't yet see integer triplets satisfying $429$. The closest seen is $432$ (if $0$ can be included $6^3+6^3+0^3=432$). For strictly positive integer triplets the nearest number is 433 ($433=6^3+6^3+1^3$).  
f[x_, y_, z_] := x^3 + y^3 + z^3
sol = Maximize[{f[x, y, z], 429 <= f[x, y, z] <= 432, x >= 0 , y >= 0,
z >= 0}, {x, y, z} \[Element] Integers]

Also, if you wish to stretch to -ve integers that is doable. For example, 
f[x_, y_, z_] := x^3 + y^3 + z^3
sol = Maximize[{f[x, y, z], 420 <= f[x, y, z] <= 430, 10 >= x >= -10 ,
10 >= y >= -10, 10 >= z >= -10}, {x, y, z} \[Element] Integers]

Will result in
     {424, {x -> -8, y -> -4, z -> 10}}
Some useful links (about recent progress): https://math.mit.edu/~drew/Waterloo2019.pdf
http://news.mit.edu/2019/answer-life-universe-and-everything-sum-three-cubes-mathematics-0910 
