Integer which is Sum of 3 Fourth Powers in 2 Ways and Products of Those Roots
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Theorem
The positive integer $256 \, 103 \, 393$ can be expressed as the sum of $3$ fourth powers in $2$ different ways:
\(\ds 256 \, 103 \, 393\) | \(=\) | \(\ds 22^4 + 93^4 + 116^4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 29^4 + 66^4 + 124^4\) |
Also note that:
\(\ds 237 \, 336\) | \(=\) | \(\ds 22 \times 93 \times 116\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 29 \times 66 \times 124\) |
Proof
We have that:
\(\ds 256 \, 103 \, 393\) | \(=\) | \(\ds 234 \, 256 + 74 \, 805 \, 201 + 181 \, 063 \, 936\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 22^4 + 93^4 + 116^4\) |
\(\ds 256 \, 103 \, 393\) | \(=\) | \(\ds 707 \, 281 + 18 \, 974 \, 736 + 236 \, 421 \, 376\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 29^4 + 66^4 + 124^4\) |
Then we have:
\(\ds 22 \times 93 \times 116\) | \(=\) | \(\ds \left({2 \times 11}\right) \times \left({3 \times 31}\right) \times \left({2^2 \times 29}\right)\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 \times 3 \times 11 \times 29 \times 31\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 237 \, 336\) |
\(\ds 29 \times 66 \times 124\) | \(=\) | \(\ds 29 \times \left({2 \times 3 \times 11}\right) \times \left({2^2 \times 31}\right)\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^3 \times 3 \times 11 \times 29 \times 31\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 237 \, 336\) |
$\blacksquare$
Historical Note
This result is reported by David Wells in his Curious and Interesting Numbers, 2nd ed. of $1997$ as the work of Stephane Vandemergel, but details are lacking.
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $256,103,393$