Results 1 to 5 of 5

Math Help - elem. # theo - "prove never perfect square"

  1. #1
    Member cassiopeia1289's Avatar
    Joined
    Aug 2007
    From
    chicago
    Posts
    101

    elem. # theo - "prove never perfect square"

    Prove 3a^2 - 1 is never a perfect square.
    The book hints to use the fact: "The square of any integer is either of the form 3k or 3k+1," which we proved in a previous problem.

    So how do you start?

    Assuming 3a^2 - 1 is a perfect square, it must be able to be written in the form of 3k or 3k+1.
    Then what? How do you show it can't or NEVER can be?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by cassiopeia1289 View Post
    Prove 3a^2 - 1 is never a perfect square.
    The book hints to use the fact: "The square of any integer is either of the form 3k or 3k+1," which we proved in a previous problem.

    So how do you start?

    Assuming 3a^2 - 1 is a perfect square, it must be able to be written in the form of 3k or 3k+1.
    Then what? How do you show it can't or NEVER can be?
    i guess you could assume a is even and show it doesn't work, then assume a is odd and show it doesn't work either. that is, you can't simplify it to get it in that form
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member cassiopeia1289's Avatar
    Joined
    Aug 2007
    From
    chicago
    Posts
    101
    Ok, so by showing that there's no possible way to put it into the form would be like "let a=2s+1" ... ect ... which is not of the form 3k or 2k+1 and so on with "let a+2s" ... ? Is that enough?

    But wait: how do we even know a is an integer at all? The problem never said so ... ?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by cassiopeia1289 View Post
    Ok, so by showing that there's no possible way to put it into the form would be like "let a=2s+1" ... ect ... which is not of the form 3k or 2k+1 and so on with "let a+2s" ... ? Is that enough?

    But wait: how do we even know a is an integer at all? The problem never said so ... ?
    yes, if you can show that, it is enough, because there are no other options. you have exhausted all possibilities. a is an integer, it has to be even or odd, and that covers all integers. so you would prove that it can never work no matter what
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by cassiopeia1289 View Post
    Assuming 3a^2 - 1 is a perfect square, it must be able to be written in the form of 3k or 3k+1.
    Because 3a^2 - 1 = 3a^2 - 3 + 2 = 3(a^2 - 1)+2.
    Therefore it has the form 3k+2 not 3k,3k+1.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: October 17th 2011, 03:50 PM
  2. Replies: 2
    Last Post: April 24th 2011, 08:01 AM
  3. Replies: 4
    Last Post: November 3rd 2010, 12:12 PM
  4. Replies: 1
    Last Post: October 25th 2010, 05:45 AM
  5. Euclid's "perfect numbers theorem"
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: October 1st 2010, 02:48 AM

Search Tags


/mathhelpforum @mathhelpforum