Results 1 to 4 of 4

Math Help - Sum of powers

  1. #1
    Member
    Joined
    May 2009
    Posts
    77

    Sum of powers

    Hi verybody,
    Is there a proof of:

    Sum[(i^s - 1), {i, 1, a}] mod( a ) = Sum[(i - 1)^s, {i, 1, a}] mod(
    a )
    a, s positive integers (could be equal).
    If there is, please tell me.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Are you saying

    \sum_{i=1}^a(i^s-1) \equiv \sum_{i=1}^a(i-1)^s \mod a

    ?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    May 2009
    Posts
    77
    exactly (a>s)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    <br />
\sum_{i=1}^a(i^s-1) \equiv \Big(\sum_{i=1}^ai^s\Big)-a \equiv  \sum_{i=1}^ai^s \mod a

    and
    \sum_{i=1}^ai^s \equiv \sum_{i=1}^a(i-1)^s \mod a

    because both sums are taken over complete residue systems \mod a. Works also if s>a b.t.w.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Powers of Two
    Posted in the Math Challenge Problems Forum
    Replies: 4
    Last Post: January 4th 2011, 04:29 AM
  2. Powers
    Posted in the Algebra Forum
    Replies: 4
    Last Post: June 6th 2010, 07:55 AM
  3. Sum of nth powers
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: May 9th 2010, 12:05 AM
  4. powers of 2
    Posted in the Algebra Forum
    Replies: 2
    Last Post: November 3rd 2008, 11:53 PM
  5. powers
    Posted in the Algebra Forum
    Replies: 6
    Last Post: February 14th 2008, 07:20 AM

Search Tags


/mathhelpforum @mathhelpforum