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It is well-know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ...$ diverges. But how about the sum below?

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Given the equation $\large 2^{n}+1=3^{k},$find all solutions where $n,\: k$ are positive integers $\leq 10^{100}$.

Enter the sum of all such $n$ and ...

When $x^9 - x$ is factorized as completely as possible into polynomials and monomials with integral coefficients, how many factors are there?

Divide $14$ into the sum of many natural numbers, and find the maximum value of the product of these numbers,

Let $x$ and $y$ be positive integers, with $x<y$, such that the following equation holds:

$\large 2xy = (x+4)(y+4)$

Find the sum of all possible values of $x$.

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