**Problem 1.**

Prove that when is a sufficiently large positive integer there exists a finite set of prime numbers such that the sum of over is equal to .

Prove that when is a sufficiently large positive integer there exists a finite set of prime numbers such that the sum of over is equal to .

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Let denote the th harmonic number . Let .Prove that for positive integers and with ,

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Find all pairs of integers such that

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Let be diffrentiable, with and with and both nowhere zero on . Let be a positive real number, and for let . Prove that is divergent.

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