Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook