Results 1 to 4 of 4

Math Help - Congruency Question

  1. #1
    Newbie
    Joined
    Sep 2010
    Posts
    3

    Congruency Question

    Hey, first post and all so maybe someone can help me out with a question from my Number Theory class. It's in the Congruences chapter.

    Prove that if n is congruent to 4 (mod 9), then n cannot be written as the sum of 3 cubes.

    Thanks in advance for the help!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor undefined's Avatar
    Joined
    Mar 2010
    From
    Chicago
    Posts
    2,340
    Awards
    1
    Quote Originally Posted by Flippinpony View Post
    Hey, first post and all so maybe someone can help me out with a question from my Number Theory class. It's in the Congruences chapter.

    Prove that if n is congruent to 4 (mod 9), then n cannot be written as the sum of 3 cubes.

    Thanks in advance for the help!
    0^3\equiv0\pmod{9}

    1^3\equiv1\pmod{9}

    2^3\equiv8\pmod{9}

    3^3\equiv0\pmod{9}

    4^3\equiv1\pmod{9}

    5^3\equiv8\pmod{9}

    6^3\equiv0\pmod{9}

    7^3\equiv1\pmod{9}

    8^3\equiv8\pmod{9}

    See where to go from here?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2010
    Posts
    3
    Ok, so what I'm seeing is that all cubes are congruent to either 0, 1, or 8 (mod 9). Therefore, a cube or a sum of cubes for that matter cannot be congruent to 4 (mod 9). Is there a theorem I should point to when writing a proof for this?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor undefined's Avatar
    Joined
    Mar 2010
    From
    Chicago
    Posts
    2,340
    Awards
    1
    Quote Originally Posted by Flippinpony View Post
    Ok, so what I'm seeing is that all cubes are congruent to either 0, 1, or 8 (mod 9). Therefore, a cube or a sum of cubes for that matter cannot be congruent to 4 (mod 9). Is there a theorem I should point to when writing a proof for this?
    That's essentially all there is, it's just an exhaustive search. Maybe there's a more elegant way, I don't know. You can list systematically and for each easily verify the sum is not 4 mod 9.

    000
    001
    008
    011
    018
    088
    111
    118
    188
    888

    Edit: I'm not sure if it's a typo when you wrote that a sum of cubes can't be congruent to 4 mod 9... of course if we have four cubes then we can just choose 1+1+1+1.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help with congruency
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: October 20th 2009, 02:41 AM
  2. New Question on congruency
    Posted in the Geometry Forum
    Replies: 1
    Last Post: September 14th 2009, 08:13 PM
  3. Congruency
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: March 7th 2009, 01:14 PM
  4. Need help with CONGRUENCY!!! @__@?
    Posted in the Geometry Forum
    Replies: 2
    Last Post: November 29th 2007, 11:59 PM
  5. Congruency
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: March 15th 2007, 08:35 PM

/mathhelpforum @mathhelpforum