After reading Arthur C. Clarke and Frederik Pohl's The Last Theorem,I thought about the problem for a while and I found that the difference between two numbers raised to the power n ( \( c^{n} = (a+b)^{n} - a^{n} \) ) can be expressed as \( \displaystyle \sum_{i=0}^n b \times a^{n-1} \times ( \frac{a+b}{a} )^{i} \).So if it can be proved that \( \displaystyle \sum_{i=0}^n b \times a^{n-1} \times ( \frac{a+b}{a} )^{i} \) cannot be equal to a number raised to the \(n\)th power then this problem could be solved.

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