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.

• You could trivially solve this by exhaustive search. If each is required to be positive then you’ll need $x,y,z \le 7$. By symmetry, you could further assume without loss of generality that $x\le y\le z$ which reduces the search space. – User8128 May 19 at 4:06
• I believe you need to delete the positive constraint in your Mathematica code. – Jaume Oliver Lafont May 19 at 4:27
• @User8128 Although the search will turn out to be short in some cases, this is not true in general. In some cases, the search is reportedly not trivial. – Rosie F May 19 at 4:57
• $182^3+284^3-307^3=429$ – Mike Bennett May 19 at 5:01

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).
• Thanks for bringing sense into this thread! – Jyrki Lahtonen May 19 at 5:17
• Nice! Needs a broader search I guess. – NivPai May 19 at 15:16
• Let me also add this link here (especially that nown=42 also has a solution). math.mit.edu/~drew/Waterloo2019.pdf – NivPai May 19 at 15:27

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}}