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- Jun 20, 2014

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Show that there are infinitely many primes of the form 4k + 1 where $k$ is an integer.

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- Thread starter
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- #1

- Jun 20, 2014

- 1,925

-----

Show that there are infinitely many primes of the form 4k + 1 where $k$ is an integer.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

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- Jun 20, 2014

- 1,925

Consider the number $N=\left(\prod{p_i}\right)^2+1$. Because $p_i\equiv1\pmod4$ for all $i$, $N\equiv2\pmod4$. As $N>2$, this shows that $N$ is not a power of $2$ and has at least one odd prime factor $q$.

We have $\left(\prod{p_i}\right)^2\equiv-1\pmod{q}$, which shows that $-1$ is a quadratic residue modulo $q$. By the laws of quadratic reciprocity, this implies that $q\equiv1\pmod4$. However, none of the $p_i$ can divide $N$, and $q$ is a prime of the form $4k+1$ different from the $p_i$; this contradicts the fact that the set $\{p_i\}$ contains all such primes.

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