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Quadratic Residues

  1. Jan 27, 2009 #1
    Q- A number is a quadratic residue modulo m if it takes the form x[tex]^{2}[/tex] mod m
    for some integer x. List the quadratic residues modulo 3, 4, 5, and 7. What
    patterns, if any, do you notice?

    modulo 7

    0^2=0, 1[tex]^{2}[/tex]=1, 2[tex]^{2}[/tex]=4, 3[tex]^{2}[/tex]=2, 4[tex]^{2}[/tex]=2, 5[tex]^{2}[/tex]=4, 6[tex]^{2}[/tex]=1

    modulo 5

    0^2=0, 1[tex]^{2}[/tex]=1, 2[tex]^{2}[/tex]=4, 3[tex]^{2}[/tex]=4, 4[tex]^{2}[/tex]=2

    modulo 4

    0^2=0, 1[tex]^{2}[/tex]=1, 2[tex]^{2}[/tex]=0, 3[tex]^{2}[/tex]=1

    modulo 3

    0^2=0, 1[tex]^{2}[/tex]=1, 2[tex]^{2}[/tex]=1

    I don't to see any patterns?
    Last edited: Jan 27, 2009
  2. jcsd
  3. Jan 27, 2009 #2


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    You forgot 0^2 = 0 in each case. But to see patterns, I think you're supposed to sort the residues and remove duplicates.

    I'm not sure what pattern you're supposed to see, actually. That primes have more residues than composites? That about half the numbers are quadratic residues mod a prime? Or is it supposed to be related to the Law of Quadratic Reciprocity that you presumably haven't learned yet?
  4. Jan 27, 2009 #3
    so if i remove duplicates then

    0^2=0, 1^2=1, 2^2=4, 3^2=2 (mod7)

    0^2=0, 1^2=1, 2^2=4, (mod5) i think 4^2=1 not 2.

    0^2=0, 1^2=1 (mod4)

    0^2=0, 1^2=1 (mod3)
  5. Jan 28, 2009 #4
    what do you mean by "That about half the numbers are quadratic residues mod a prime?"
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