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

  • Thread starter Vespero
  • Start date
  • #1
28
0

Homework Statement



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


Homework 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]


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:

Answers and Replies

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

Replies
12
Views
2K
Replies
12
Views
2K
Replies
3
Views
2K
Replies
2
Views
924
Replies
2
Views
1K
  • Last Post
Replies
1
Views
811
Replies
6
Views
2K
Top