Results 1 to 4 of 4

Math Help - Quadratic Residues

  1. #1
    Newbie
    Joined
    Nov 2012
    From
    champaign
    Posts
    5

    Quadratic Residues

    Let p be a prime number with p>3. Prove that the sum of the quadratic residues modulo p is divisible by p.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,054
    Thanks
    368
    Awards
    1

    Re: Quadratic Residues

    Quote Originally Posted by babygill View Post
    Let p be a prime number with p>3. Prove that the sum of the quadratic residues modulo p is divisible by p.
    How far have you gotten with this?

    -Dan
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2012
    From
    champaign
    Posts
    5

    Re: Quadratic Residues

    I don't know where to start
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,054
    Thanks
    368
    Awards
    1

    Re: Quadratic Residues

    Okay. I've refreshed my memory of this topic, but I am in no way a professional.

    First, I am thinking that your problem statement is incorrect. (Which may mean that I'm incorrect, but anyway...)

    For example, take a look at the multiplicative group mod 7: \mathBB{Z} / _7 \mathbb{Z} = \{ 1, 2, 3, 4, 5, 6 \} .

    Squaring each of these elements gives the quadratic residues by inspection: q = 1, 4. The sum of these is obviously not divisible by 7. I think what you are looking for is the sum of the square of all members of the group. This can be easily shown by looking at the summation:
    \sum_{i = 1}^6~i^2

    This turns out to be divisible by 7 as required. (And thus the sum of the residues is also divisible by 7. Prove this!) I leave it to you to look at the general case.

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: October 12th 2012, 01:01 AM
  2. quadratic residues
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: April 16th 2012, 03:22 AM
  3. Quadratic Residues
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: July 23rd 2009, 11:25 AM
  4. Quadratic Residues
    Posted in the Math Challenge Problems Forum
    Replies: 1
    Last Post: July 17th 2009, 08:26 PM
  5. Sum of quadratic residues
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: October 27th 2008, 06:45 PM

Search Tags


/mathhelpforum @mathhelpforum