Dedicated to \(e^{i\pi} + 1729\)

\[\left(\sum_{n=1}^{10000} 2S_m + 5S_n\right)^{1729}\]

Let \(S_n\) and \(S_m\) be two natural numbers for arbitary natural numbers \(n\) and \(m\) : such that , \(S_{n-1}=2S_{m+1}\).Also \(5^m | S_m\) , provided that \((n,m) \leq 10000\).

Then evaluate above sumation modulo 10

Bonus : Generalize this.

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