Results 1 to 2 of 2

Thread: Set of residues

  1. #1
    Super Member Showcase_22's Avatar
    Sep 2006
    The raggedy edge.

    Set of residues

    Is there a complete set of residues modulo p consisting of perfect squares:

    i). When p=7.

    ii). for any prime p.
    I can probably work out part ii). if I can do part i).

    I think part i) is true but I can't figure out the set. My set so far is $\displaystyle \{0,36,9\}$ for 0,1 and 2 respectively and $\displaystyle \{25 \}$ for 4.

    I can't find any perfect squares for 3,5 and 6!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Apr 2009
    Atlanta, GA

    There is no such set ever

    Consider the sequence of squares $\displaystyle 0,1,4,9,16,25,36,49,64,81,100,...$ in modulo:

    $\displaystyle \bmod 7: 0,1,4,2,2,4,1,0,1,4,2,2,4,1,0,1,...$ (notice a pattern?)

    Notice that $\displaystyle (n+7)^2-n^2=14n+7 \equiv 0 (\bmod 7)$, so the set repeats in blocks of $\displaystyle 7$.

    Also, $\displaystyle (7k+r)^2-(7k-r)^2=28kr \equiv 0 (\bmod 7)$ so the set "reflects" as shown in this example. More clearly, for any $\displaystyle k$, $\displaystyle (7k)^2 \equiv 0 (\bmod 7)$, therefore $\displaystyle (7k+r)^2 \equiv (7k-r)^2 (mod 7)$ for all $\displaystyle r$.

    Both of these observations can be generalized. For any prime $\displaystyle p$,

    $\displaystyle (n+p)^2-n^2 \equiv 0 (\bmod p)$

    $\displaystyle (pk+r)^2-(pk-r)^2 \equiv 0 (\bmod p)$

    For all appropriate $\displaystyle n,k,r$

    Therefore, the size of the set of residues of $\displaystyle n^2$ for any prime $\displaystyle p$ is at most $\displaystyle \lceil \frac{p}{2} \rceil$

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. residues
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: May 7th 2010, 05:18 AM
  2. About sum of residues
    Posted in the Number Theory Forum
    Replies: 8
    Last Post: May 18th 2009, 11:54 AM
  3. residues
    Posted in the Calculus Forum
    Replies: 6
    Last Post: Feb 11th 2009, 08:44 AM
  4. residues 2
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Feb 10th 2009, 08:19 AM
  5. Residues
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Nov 6th 2008, 09:29 AM

Search Tags

/mathhelpforum @mathhelpforum