Results 1 to 6 of 6

Math Help - difference of two squares

  1. #1
    perfect square
    Guest

    difference of two squares

    Find two integers whose squares have a difference of 1,234,567.





    can you find two integers whose squares will produce a difference of any whole number you chose?


    if so, how? if not, what kind of numbers can't be chosen.













    well... a^2-b^2=1234567




    we know a must end in a 1 or 9 and b must end in a 2 or 8 (this is to produce the seven at the end)





    i came up with 617289^2-617288^2=1234577 and that's as close as i can get.





    any suggestions? first time poster! sorry if it is an unusual / inappropriate post!







    much thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    n^{2}-(n-1)^{2}=1234567

    n=617284

    617284^{2}-617283^{2}=1234567
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Number suchs that,
    n\equiv 0,1,3 (\bmod 4)
    Are expressible as a difference of two square.

    And,
    n\equiv 2 (\bmod 4)
    Are not.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,811
    Thanks
    701
    Hello, perfect square!

    Find two integers whose squares have a difference of 1,234,567
    Galactus used a very clever bit of trivia.

    Consecutive squares differ by consecutive odd numbers.

    . . \begin{array}{cccc} & & & \text{diff} \\ 0^2 & = & 0 & \\ & & & 1 \\ 1^2 & = & 1 & \\ & & & 3 \\ 2^2 & = & 4 & \\ & & & 5 \\ 3^2 & = & 9 & \\ & & & 7 \end{array}
    . . \begin{array}{cccc}4^2 & = & 16 & \\ & & & 9 \\ 5^2 & = & 25 & \\ \vdots & & \vdots & \vdots \end{array}


    As he pointed out, we want two consecutive integers with a difference of 1234567.

    So we have: . (n + 1)^2 - n^2 \:=\:1,234,567

    . . n^2 + 2n + 1 - n^2 \:=\:1,234,567\quad\Rightarrow\quad 2n+1 \:=\:1,234,567\quad\Rightarrow\quad 2n \:=\:1,234,566

    . . Hence: . n \,=\,617,283

    Therefore: . 617,284^2 - 617283^2 \:=\:1,234,567

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    I suspect that 1,234,567 is prime (but I'm not sure).

    If the difference is a composite odd number,
    . . there may be an alternate solution.


    Example: Find two integers whose squares differ by 133.

    Since \frac{133-1}{2} = 66, we have: . 67^2 - 66^2 \:=\:133


    . Since 133 = 7\times 19, we have: . . 19 + 19 + 19 + 19 + 19 + 19 + 19
    . . . . . which can be written: . . 13 + 15 + 17 + 19 + 21 + 23 + 25
    which are the differences of: . 6^2\quad7^2\quad\:8^2\quad\:9^2\quad10^2\quad11^2\  quad12^2\quad13^2

    Therefore: . 13^2 - 6^2 \:=\:133

    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member DivideBy0's Avatar
    Joined
    Mar 2007
    From
    Melbourne, Australia
    Posts
    432
    Quote Originally Posted by Soroban View Post
    So we have: . (n + 1)^2 - n^2 \:=\:1,234,567

    . . n^2 + 2n + 1 - n^2 \:=\:1,234,567\quad\Rightarrow\quad 2n+1 \:=\:1,234,567\quad\Rightarrow\quad 2n \:=\:1,234,566

    . . Hence: . n \,=\,617,283
    Such a seemingly difficult question, transformed into simple algebra! Once again! Nice!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    I suspect that 1,234,567 is prime (but I'm not sure).

    No, Soroban, it is not prime.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Difference of Squares
    Posted in the Algebra Forum
    Replies: 1
    Last Post: March 31st 2011, 04:49 PM
  2. Difference of squares
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: March 31st 2010, 10:41 AM
  3. Ti-83 difference of squares
    Posted in the Calculators Forum
    Replies: 4
    Last Post: August 4th 2009, 09:52 PM
  4. Difference of Squares
    Posted in the Algebra Forum
    Replies: 9
    Last Post: August 25th 2008, 09:28 AM
  5. The difference of two squares
    Posted in the Math Topics Forum
    Replies: 2
    Last Post: August 8th 2008, 11:59 AM

/mathhelpforum @mathhelpforum