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