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), 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 ...

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Calculate so that

Where 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 is divergent.