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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.

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  • $\begingroup$ 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. $\endgroup$ – User8128 May 19 at 4:06
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    $\begingroup$ I believe you need to delete the positive constraint in your Mathematica code. $\endgroup$ – Jaume Oliver Lafont May 19 at 4:27
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    $\begingroup$ @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. $\endgroup$ – Rosie F May 19 at 4:57
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    $\begingroup$ $182^3+284^3-307^3=429$ $\endgroup$ – Mike Bennett May 19 at 5:01
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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).
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    $\begingroup$ Thanks for bringing sense into this thread! $\endgroup$ – Jyrki Lahtonen May 19 at 5:17
  • $\begingroup$ Nice! Needs a broader search I guess. $\endgroup$ – NivPai May 19 at 15:16
  • $\begingroup$ Let me also add this link here (especially that nown=42 also has a solution). math.mit.edu/~drew/Waterloo2019.pdf $\endgroup$ – NivPai May 19 at 15:27
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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

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  • $\begingroup$ And the closest to 627 is 622=7^3+7^3-4^3 $\endgroup$ – zeraoulia rafik May 19 at 4:34
  • $\begingroup$ if the solution really exist in your given range , I think we should retire wolfram alpha $\endgroup$ – zeraoulia rafik May 19 at 4:46
  • $\begingroup$ I checked with Mathematica. It works. $\endgroup$ – NivPai May 19 at 4:48

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