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Proof of infinite primes

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Prove: There are infinitely many prime numbers p of the form 4n+3. In other words, show that there exist infinitely many positive integers, n, such that the number 4n+1 is prime.

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Solution Summary

This is a proof regarding infinite numbers of primes.

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Note: the notation "p_k" means p with a subscript k.

The proof of this problem can be seen as a kind of generalization of Euclid's proof that there are infinitely many primes. You have most likely seen Euclid's proof if you are in a basic undergraduate number theory course.

The solution of this problem, like Euclid's, is a proof by contradiction:

Assume there are only finitely many primes of the form 4n+3.
Let ...

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