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If $p$ and $q$ are coprime positive integers s.t. $\frac{p}{q}=\sum_{k=0}^{100}\frac1{3^{2^k}+1}$, what is the smallest prime factor of $p$?
If the sum $$S=\frac14+\frac1{10}+\frac1{82}+\frac1{6562}+\cdots+\frac1{3^{2^{100}}+1}$$
is expressed in the form $\frac pq,$ where $p,q\in\mathbb N$ and $\gcd(p, q) =1.$ Then what is smallest prime ...
