This is bugging me for quite some time now, any hints are appreciated:

Prove: Given any finite sequence of prime numbers which are equivalent to 3 (mod 4), $\displaystyle (p_0,p_1,...,p_k)$ you can construct a new prime number which is equivalent to 3 (mod 4).

Attempt: I immediately thought about Euclid's proof of the infinity of the collection of primenumbers. So maybe we should multiply the given primenumbers, which gives us a number that is equivalent to $\displaystyle 3^k (mod 4)$...

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Calculate $\displaystyle n$ so that $\displaystyle \sum_{p \in \mathbb{P},p\leq n}\frac{1}{p}>5$

Where $\displaystyle \mathbb{P}$ represents the prime numbers.

And of course I want the first n for which this is true. If any more theory is needed for the second question I can put it here, but it is vaguely written so I don't really get it. It is a constructive proof by Euler that shows that $\displaystyle \sum_{n\in \mathbb{P}\frac{1}{n}$ is divergent.