# Talk:492 is Sum of 3 Cubes in 3 Ways

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I think Wells confused this result with the famous sum of 3 cubes problem.

Evidence 1: as of 2009, for $\size x, \size y, \size z < 10^{14}$, $x^3 + y^3 + z^3 = 492$ has 3 solutions.

Source: *New Sums of Three Cubes*, by Elsenhans.

\(\displaystyle 492\) | \(=\) | \(\displaystyle 50^3 + \paren {-19}^3 + \paren {-49}^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 123134^3 + 9179^3 + \paren {-123151}^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1793337644^3 + \paren {-813701167}^3 + \paren {-1735662109}^3\) |

- I just checked that report. I presume it's this one: https://www.ams.org/journals/mcom/2009-78-266/S0025-5718-08-02168-6/S0025-5718-08-02168-6.pdf

- There doesn't seem to be anything in it about $492$, am I looking in the wrong place? --prime mover (talk) 14:32, 21 July 2020 (UTC)

- Here. https://www.uni-math.gwdg.de/jahnel/Arbeiten/Liste/threecubes_20070419.txt --RandomUndergrad (talk) 14:38, 21 July 2020 (UTC)

- Thank you for that, couldn't find it when I searched, it was very well hidden on Jahnel's page. --prime mover (talk) 14:59, 21 July 2020 (UTC)

- Interestingly, $492$ stands out as having so few representations as sums of cubes, most of the other numbers have many more. I have work to do this evening, then. --prime mover (talk) 14:59, 21 July 2020 (UTC)

Evidence 2: In *Mathematics on Vacation*, Madachy says:

- "The American Mathematical Monthly of January 1957 asked for an integer less than $1000$ whose cube could be represented in 5 distinct ways as the sum of 3 positive integers."

The exceptional solutions given were $492, 792, 870$, with $10, 11, 10$ ways to represent them.

In particular:

\(\displaystyle 492^3\) | \(=\) | \(\displaystyle 24^3 + 204^3 + 480^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 48^3 + 85^3 + 491^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 72^3 + 384^3 + 396^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 113^3 + 264^3 + 463^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 114^3 + 360^3 + 414^3\) | In Madachy's list but misprinted $114$ as $144$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 149^3 + 336^3 + 427^3\) | not in Madachy's list | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 176^3 + 204^3 + 472^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 190^3 + 279^3 + 449^3\) | not in Madachy's list | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 207^3 + 297^3 + 438^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 226^3 + 332^3 + 414^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 243^3 + 358^3 + 389^3\) | not in Madachy's list | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 246^3 + 328^3 + 410^3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 281^3 + 322^3 + 399^3\) |

The above 13 representations were given by my program (in 1 second). Excluding the 3 he missed, there are 10 in the book.

Madachy credited this list to David A. Klarner of Edmonton, Alberta.

For those interested all the numbers above have 13 representations, and according to A316359 on OEIS, $492$ is the smallest such number. --RandomUndergrad (talk) 13:47, 21 July 2020 (UTC)

- Yes I've found it now, I looked it up in my own copy of that Madachy book. I'll process this page accordingly. --prime mover (talk) 14:22, 21 July 2020 (UTC)

- Please feel free to review the pages that I have created based on the above research. --prime mover (talk) 21:39, 21 July 2020 (UTC)