Positive Integer Sum of 3 Fourth Powers in 2 Ways
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Theorem
The smallest positive integer which can be expressed as the sum of $3$ fourth powers in $2$ different ways is $6578$:
| \(\ds 1^4 + 2^4 + 9^4\) | \(=\) | \(\ds 1 + 16 + 6561\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 6578\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 81 + 2401 + 4096\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3^4 + 7^4 + 8^4\) |
Proof
The fact that this is the smallest can be demonstrated by calculation.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $6578$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $6578$