492 Cubed is Sum of 3 Positive Cubes in 13 Ways

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Theorem

The cube of $492$ can be expressed as the sum of $3$ positive cubes in $13$ different ways:

\(\ds 492^3\) \(=\) \(\ds 24^3 + 204^3 + 480^3\)
\(\ds \) \(=\) \(\ds 48^3 + 85^3 + 491^3\)
\(\ds \) \(=\) \(\ds 72^3 + 384^3 + 396^3\)
\(\ds \) \(=\) \(\ds 113^3 + 264^3 + 463^3\)
\(\ds \) \(=\) \(\ds 114^3 + 360^3 + 414^3\)
\(\ds \) \(=\) \(\ds 149^3 + 336^3 + 427^3\)
\(\ds \) \(=\) \(\ds 176^3 + 204^3 + 472^3\)
\(\ds \) \(=\) \(\ds 190^3 + 279^3 + 449^3\)
\(\ds \) \(=\) \(\ds 207^3 + 297^3 + 438^3\)
\(\ds \) \(=\) \(\ds 226^3 + 332^3 + 414^3\)
\(\ds \) \(=\) \(\ds 243^3 + 358^3 + 389^3\)
\(\ds \) \(=\) \(\ds 246^3 + 328^3 + 410^3\)
\(\ds \) \(=\) \(\ds 281^3 + 322^3 + 399^3\)


Proof

Brute force.


Also see


Historical Note

The question was asked in American Mathematical Monthly of January $1957$ for a (positive) integer less than $1000$ whose cube could be expressed as the sum of $3$ cubes in $5$ distinct ways.

A solution submitted by Leon Bankoff returned some $47$ such integers which have between $5$ and $9$ such expressions, but surprisingly $492$ was not among them.

$870$ had most such representations, that is, $9$.

Subsequent investigation unearthed several more.

By the time Joseph S. Madachy reported in his Mathematics on Vacation of $1966$, there were $2$ integers each with $10$ such representations (that is, $492$ and $870$), and one had $11$ (that is, $792$).

The representations for $492$ and $792$ were found by David A. Klarner, while the representations for $870$ were, as for the $1957$ investigation, attributed to Leon Bankoff.


A more recent investigation, using a computer program designed to test exhaustively all integers less than $1000$ has revealed that each of $492$, $792$ and $870$ has exactly $13$ such representations.


Sources

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