492 is Sum of 3 Cubes in 3 Ways/Mistake

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Source Work

1986: David Wells: Curious and Interesting Numbers:

The Dictionary
$492$


Mistake

$492$ is the sum of $3$ cubes, one or two of which may be negative, in no fewer than $10$ different ways. [Madachy]


Correction

David Wells appears to have conflated $2$ separate results.


The number of known ways $492$ can be expressed as the sum of $3$ cubes, either positive or negative, is $3$:

\(\ds 492\) \(=\) \(\ds 50^3 + \paren {-19}^3 + \paren {-49}^3\)
\(\ds \) \(=\) \(\ds 123 \, 134^3 + 9179^3 + \paren {-123 \, 151}^3\)
\(\ds \) \(=\) \(\ds 1 \, 793 \, 337 \, 644^3 + \paren {-813 \, 701 \, 167}^3 + \paren {-1 \, 735 \, 662 \, 109}^3\)


The result given in Joseph S. Madachy's $1966$ work Mathematics on Vacation is that it is possible to express $492^3$ in no fewer than $10$ different ways:

\(\ds 492^3\) \(=\) \(\ds 24^3 + 204^3 + 480^3\)
\(\ds \) \(=\) \(\ds 48^3 + 85^3 + 491^3\)
\(\ds \) \(=\) \(\ds 72^3 + 384^3 + 396^3\)
\(\ds \) \(=\) \(\ds 113^3 + 264^3 + 463^3\)
\(\ds \) \(=\) \(\ds 144^3 + 360^3 + 414^3\)
\(\ds \) \(=\) \(\ds 176^3 + 204^3 + 472^3\)
\(\ds \) \(=\) \(\ds 207^3 + 297^3 + 438^3\)
\(\ds \) \(=\) \(\ds 226^3 + 332^3 + 414^3\)
\(\ds \) \(=\) \(\ds 246^3 + 328^3 + 410^3\)
\(\ds \) \(=\) \(\ds 281^3 + 322^3 + 399^3\)

(except that the $5$th expression is incorrect: $144$ should read $114$).


Three further such expressions for $492^3$ have since been found:

\(\ds 492^3\) \(=\) \(\ds 149^3 + 336^3 + 427^3\)
\(\ds \) \(=\) \(\ds 190^3 + 279^3 + 449^3\)
\(\ds \) \(=\) \(\ds 243^3 + 358^3 + 389^3\)

$492$ is not the only integer whose cube can be expressed as the sum of $3$ positive cubes in $13$ ways, but it is the smallest.


This entry has been removed from 1997: David Wells: Curious and Interesting Numbers (2nd ed.).


Sources

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