What is the solution of $x^3+y^3+z^3=429$ in integers?

Question

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.

Answer 1

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).

Answer 2

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