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

  1. Nov 1, 2011 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations

    [itex]\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} [/itex]

    [itex]\left(\frac{ab}{p}\right) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right)[/itex]

    3. 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 [itex]p_1, p_2, ..., p_k[/itex]. I then formed the number [itex](p_{1}p_{2}...p_{k})^{2} + 1[/itex]. This number is not divisible by any [itex]p_{i}, 1 \leq i \leq k[/itex], for otherwise we would have [itex]p_{i}|1[/itex]. Now, this number has a prime factorization by the Fundamental Theorem of Arithmetic as [itex](p_{1}p_{2}...p_{k})^{2} + 1 = q_{1}q_{2}...q_{r}[/itex]. 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.
    Last edited: Nov 1, 2011
  2. jcsd
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