# Prove infinitude of primes of form 4k+1 using properties of Legendre symbol (-1/p)

## Homework Statement

Show that there are infinitely many primes $4k+1$ using the properties of $\left(\frac{-1}{p}\right)$.

## Homework Equations

$\left(\frac{-1}{p}\right) = \begin{cases} 1, & \text{if }p\equiv 1\ (mod\ 4), \\ -1, & \text{if }p\equiv 3\ (mod\ 4). \end{cases}$

$\left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right)$

## The Attempt at a Solution

Similar to other proofs involving the infinitude of the primes, I assumed that there were only a finite number of primes of the form 4k+1, specifically $p_1, p_2, ..., p_k$. I then formed the number $(p_{1}p_{2}...p_{k})^{2} + 1$. This number is not divisible by any $p_{i}, 1 \leq i \leq k$, for otherwise we would have $p_{i}|1$. Now, this number has a prime factorization by the Fundamental Theorem of Arithmetic as $(p_{1}p_{2}...p_{k})^{2} + 1 = q_{1}q_{2}...q_{r}$. From this point on, I'm not sure where to go. Clearly, I need to incorporate the Legendre symbol, but I can't see where to apply its properties to arrive at a contradiction.

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