Results 1 to 7 of 7

Math Help - Perfect Square Test

  1. #1
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943

    Perfect Square Test

    Are there any tests you can perform to see whether an integer is a perfect square? The number I want to test is odd. The only test I know is: If k is odd, then k^2 \equiv 1 \mod 8. Unfortunately, the number I have passes this test. Are there any others I can do?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Aug 2009
    Posts
    170
    Thanks
    8
    Um, from a previously answered question, you can also check the number of divisors it has . If it's odd, then it's a perfect square, if it's even then it's not.

    Not sure if that's the kind of test you're looking for, but it works
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    I don't know the number, though. I'll describe the problem below, but I DO NOT WANT THE ANSWER OR EVEN A SOLUTION; I just want some tests that I can perform and maybe a nudge in the right direction.

    Problem:

    Can a number of the form 200...009 (i.e. 2\cdot 10^n + 9, n\in\mathbb{N}) be a perfect square?

    Any potential square root has to be of the form 10j+3 or 10k+7. Squaring the first option gives 100j^2+60j+9=20..09 \implies j(5j+3) = 10^m, m\in\mathbb{N}. I think with a bit of work I could show that there are no integer solutions to that equation; however I'm not so sure I can do the same for (10k+7)^2=20..09.

    Does this seem like a good way to go about this problem, or should I try something completely different?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jan 2009
    Posts
    591
    Quote Originally Posted by redsoxfan325 View Post
    Are there any tests you can perform to see whether an integer is a perfect square? The number I want to test is odd. The only test I know is: If k is odd, then k^2 \equiv 1 \mod 8. Unfortunately, the number I have passes this test. Are there any others I can do?
    I do not understand the question.
    What is wrong with extracting the Square Root?

    If the square root is an integer, it is a perfect square.
    That just seems to be the most efficient way to get the result.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    I can't do that because I don't know the number. (I explain the problem in detail in my second post.) I know that it starts with 2, ends with 9, and has an arbitrary number of zeroes in between. The question: can a number like that ever be a perfect square (for any number of zeroes).
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    You're looking too far.

    Hint : any integer is congruent mod 3 to the sum of its decimal digits.

    Now what are the squares, mod 3?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    Joined
    Jan 2009
    Posts
    591
    Quote Originally Posted by Bruno J. View Post
    You're looking too far.
    Hint : any integer is congruent mod 3 to the sum of its decimal digits.
    Now what are the squares, mod 3?
    Brilliant!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. perfect square trinomial/square of binomial
    Posted in the Algebra Forum
    Replies: 2
    Last Post: March 3rd 2011, 04:02 PM
  2. Replies: 1
    Last Post: July 21st 2010, 02:24 PM
  3. Perfect square
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: September 13th 2009, 08:34 AM
  4. Perfect square
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: October 19th 2008, 08:05 PM
  5. perfect square
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: April 21st 2007, 09:37 PM

Search Tags


/mathhelpforum @mathhelpforum