Smallest Integer which is Sum of 3 Fifth Powers in 2 Ways

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Theorem

The smallest positive integer which can be expressed as the sum of $3$ fifth powers in $2$ different ways:

The positive integer $1 \, 375 \, 298 \, 099$ can be expressed as the sum of $3$ fifth powers in $2$ different ways:

\(\ds 1 \, 375 \, 298 \, 099\) \(=\) \(\ds 24^5 + 28^5 + 67^5\)
\(\ds \) \(=\) \(\ds 3^5 + 54^5 + 62^5\)


Proof

\(\ds 1 \, 375 \, 298 \, 099\) \(=\) \(\ds 7 \, 962 \, 624 + 17 \, 210 \, 368 + 1 \, 350 \, 125 \, 107\)
\(\ds \) \(=\) \(\ds 24^5 + 28^5 + 67^5\)


\(\ds 1 \, 375 \, 298 \, 099\) \(=\) \(\ds 243 + 459 \, 165 \, 024 + 916 \, 132 \, 832\)
\(\ds \) \(=\) \(\ds 3^5 + 54^5 + 62^5\)


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Historical Note

David Wells, in his $1986$ work Curious and Interesting Numbers, attributes this result to Ronald Alter, but gives no details.


Sources

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