Smallest 5th Power equal to Sum of 5 other 5th Powers

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Theorem

The smallest positive integer whose fifth power can be expressed as the sum of $5$ other distinct positive fifth powers is $72$:

$72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5$


Proof

\(\ds 19^5 + 43^5 + 46^5 + 47^5 + 67^5\) \(=\) \(\ds 2 \, 476 \, 099\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 147 \, 008 \, 443\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 205 \, 962 \, 976\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 229 \, 345 \, 007\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 1 \, 350 \, 125 \, 107\)
\(\ds \) \(=\) \(\ds 1 \, 934 \, 917 \, 632\)
\(\ds \) \(=\) \(\ds 72^5\)


Let $\tuple {I, J, K, L, M}$ be the $5$ distinct positive fifth powers where $I < J < K < L < M$

The following program in R iterates through 97,330,464 calculations and verifies the assertion. 


for (I in 1:19) {
	for (J in 2:43) {
		for (K in 3:46) {
			for (L in 4:47) {
				for (M in 5:67) {
					if (abs((I^5+J^5+K^5+L^5+M^5)^0.2 - round((I^5+J^5+K^5+L^5+M^5)^0.2)) < 0.00000001) {
						print(paste((I^5+J^5+K^5+L^5+M^5)^0.2, I, J, K, L, M)) 
					}
				}
			}
		}
	}
}


Sources

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