show-for-all-n-N-that-3-1-5-n-5-is-divisible-by-1-3-n-3-

Question Number 192233 by gatocomcirrose last updated on 12/May/23

show for all n \in N that

3 (1^{5} + \dots + n^{5}) is divisible by 1^{3} + \dots + n^{3}

Commented by Frix last updated on 12/May/23

S_{5} = \underset{j = 1}{\sum^{n}} j^{5} = \frac{n^{2} {(n + 1)}^{2} (2 n^{2} + 2 n - 1)}{12}

S_{3} = \underset{j = 1}{\sum^{n}} j^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}

\frac{3 S_{5}}{S_{3}} = 2 n^{2} + 2 n - 1

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