Results 1 to 2 of 2

Thread: Sums of squares of positive integers prime to n

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    14

    Sums of squares of positive integers prime to n

    Let $\displaystyle S(n)$ denote the sum of the squares of the positive integers $\displaystyle \leq n$ and prime to $\displaystyle n$.
    Prove that
    $\displaystyle \sum_{j = 1}^n{j^2} = \sum_{d \mid n}{d^2S\left(\frac{n}{d}\right)} = \sum_{d \mid n}{\frac{n^2}{d^2}S(d)}$

    I have trouble with separating the integers $\displaystyle \leq n$ into classes, so that all integers $\displaystyle k$ such that $\displaystyle (k,n) = d$ are in the same class.
    Thanks for the help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374
    Let $\displaystyle N=\{1^2,2^2,....n^2\}$, for each d|n, define $\displaystyle N_d=\{x^2x,n)=d\}$, then $\displaystyle \{N_d:d|n\}$ is a partion on N.
    And for each $\displaystyle N_d$, Since $\displaystyle (\frac{x}{d},\frac{n}{d})=1$, By the definition of S(.), the sum of elements in $\displaystyle N_d$ is $\displaystyle d^2\times S(\frac{n}{d})$,
    Thus,the first equality of$\displaystyle
    \sum_{j = 1}^n{j^2} = \sum_{d \mid n}{d^2S\left(\frac{n}{d}\right)} = \sum_{d \mid n}{\frac{n^2}{d^2}S(d)}$
    is proved.
    When d exhaust all the divisors of n, so do $\displaystyle \frac{n}{d}$, Thus the second equality obviously hold.
    Last edited by Shanks; Nov 24th 2009 at 02:10 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Positive integers that are differences of squares
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: Jul 2nd 2011, 07:11 PM
  2. Replies: 2
    Last Post: Mar 1st 2011, 10:54 AM
  3. Replies: 2
    Last Post: Jul 18th 2010, 02:57 PM
  4. Sums of squares
    Posted in the Number Theory Forum
    Replies: 13
    Last Post: Jan 7th 2010, 03:17 PM
  5. Replies: 2
    Last Post: Oct 25th 2008, 03:19 PM

Search Tags


/mathhelpforum @mathhelpforum