Results 1 to 4 of 4

Math Help - Sums of squares : putnam problem

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

    Sums of squares : putnam problem

    Hello! This is a nice problem from Putnam a few years ago, for fun.

    Show that there are infinitely many triples of consecutive integers, each of which is the sum of two squares.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Bruno J. View Post
    Hello! This is a nice problem from Putnam a few years ago, for fun.

    Show that there are infinitely many triples of consecutive integers, each of which is the sum of two squares.
    let n=4k^2(k^2+1), \ k \in \mathbb{Z}. then n=(2k^2)^2 + (2k)^2, \ n+1=(2k^2 + 1)^2, \ n+2=(2k^2+1)^2 + 1.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Very nice! Your intuition is very good.

    My solution is different. I show that given any triple n-1, n, n+1, you can construct another triple. Since sums of two squares are closed under product, (n-1)(n+1)=n^2-1 is a sum of two squares, and thus we obtain n^2-1,n^2,n^2+1. So it suffices to show that we have one such triple, and 8,9,10 does the job.

    Yours is better though
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Bruno J. View Post
    Very nice! Your intuition is very good.

    My solution is different. I show that given any triple n-1, n, n+1, you can construct another triple. Since sums of two squares are closed under product, (n-1)(n+1)=n^2-1 is a sum of two squares, and thus we obtain n^2-1,n^2,n^2+1. So it suffices to show that we have one such triple, and 8,9,10 does the job.

    Yours is better though
    your solution is pretty good too!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Fermat's theorem on sums of two squares
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: August 2nd 2010, 01:33 PM
  2. Sums of squares
    Posted in the Number Theory Forum
    Replies: 13
    Last Post: January 7th 2010, 04:17 PM
  3. Sums of squares exercise
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: December 19th 2009, 02:15 PM
  4. Sums of squares of positive integers prime to n
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: November 24th 2009, 12:15 AM
  5. Sums of Squares problem.
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: April 7th 2008, 02:22 AM

Search Tags


/mathhelpforum @mathhelpforum