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Infinite primes using Quadratic Residues

  1. Dec 11, 2005 #1
    I've been able to prove that the set {8n+7} has infinite primes by manipulating Fermat's Theorem, but I'm trying to reprove it using quadratic residue and Legendre Polynomials.

    I've been able to show that for p=8n+7, (2/p)=1 and (-1,p)=-1

    So it follows that (-2/p)=-1. And that (-2/p)=1 iff p congruent to 1 or 5 mod 8.

    any ideas how to extend that to the final proof?
  2. jcsd
  3. Dec 12, 2005 #2


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    You mean legendre symbols.

    "And that (-2/p)=1 iff p congruent to 1 or 5 mod 8."

    is false, (-2/5)=(3/5)=-1.

    Given a finite set of primes congruent to 7 mod 8, can you construct a number whose odd prime divisors exlclude two of the residue classes {1,3,5} mod 8? This is your first step. This is similar to proving infinitely many primes of the form 4k+1. You constructed a number that has no prime divisors congruent to 3 mod 4, which you prove via quadratic residues. The 7 mod 8 case is more work, you won't be able to exclude all the other residue classes mod 8 but this is a start.
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