492 is Sum of 3 Cubes in 3 Ways
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Theorem
$492$ can be expressed as the sum of $3$ cubes, either positive or negative in $3$ known ways.
\(\ds 492\) | \(=\) | \(\ds 50^3 + \paren {-19}^3 + \paren {-49}^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 123 \, 134^3 + 9179^3 + \paren {-123 \, 151}^3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \, 793 \, 337 \, 644^3 + \paren {-81 \, 3701 \, 167}^3 + \paren {-1 \, 735 \, 662 \, 109}^3\) |
Proof
Brute force.
Also see
Historical Note
Andreas-Stephan Elsenhans and Jörg Jahnel reported in $2009$ on a systematic investigation they performed on all the solutions to the equation:
- $x^3 + y^3 + z^3 = n$
for all $0 < n < 1000$ and such that $\size x, \size y, \size z \le 10^{14}$.
Within that range, they discovered that there are exactly $3$ solutions for $n = 492$.
Of all the numbers from $0$ to $1000$, most have many more than $3$ such ways, although a few still have no known solutions.
The full results can be found at https://www.uni-math.gwdg.de/jahnel/Arbeiten/Liste/threecubes_20070419.txt.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $492$