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We know that there are infinite number of primes so as there are infinite number of primes of the form $4n+3$ where $n\in Z^+$.

A note on Burton's book (Elementary Number Theory) somehow says that it is of high chance to expect that there are also infinite number of primes of the form $4n+1$. However it is not yet proven by the time the book was published.

A quick search on net gives infinitude of primes of the form $3n+1$ and $5n+1$.

The question is: The problem: Are there infinitely many number of primes of the form $4n+1$ still an open problem or it was already proven? If so, where can I find the proof? Thanks a lot.

marked as duplicate by Travis, Mark Bennet, user133281, Claude Leibovici, Mark Fantini Jan 14 '15 at 8:38

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  • 1
    Dirichlet's Theorem shows that there are infinitely many primes of the form $a n + b$ for coprime $a, b$:… – Travis Jan 14 '15 at 7:49
  • Thanks a lot for the information. – Jr Antalan Jan 14 '15 at 23:23
  • You're welcome. – Travis Jan 14 '15 at 23:25
  • @Travis Very nice pointer. Wish, if you could please see my post that explores the subtleties of the proof of this OP's question. – jiten Jan 27 at 15:23
up vote 1 down vote accepted

Hint: Suppose there are only finitely many such primes of the form $4n+1$, call them: $p_1, p_2,\cdots, p_n$. Its an exercise for you to "show" that: $p_{n+1} = 4(p_1p_2\cdots p_n)^2 + 1$ is a prime also of the form $4n+1$.

  • Thank you so much, it helps a lot. – Jr Antalan Jan 14 '15 at 23:23

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