Results 1 to 5 of 5

Math Help - a perfect square

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    35

    a perfect square

    Let n=pq and q-p=2d where d>0 and p, q are odd prime numbers.
    1-Show that n+d2 is a perfect square i.e. n+d2=m2 for some integer m.
    2-If you know that n+d2 is a perfect square, can you find prime factor of n
    how can I show that, any help will be apprciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by koko2009 View Post
    Let n=pq and q-p=2d where d>0 and p, q are odd prime numbers.


    1-Show that n+d2 is a perfect square i.e. n+d2=m2 for some integer m.
    2-If you know that n+d2 is a perfect square, can you find prime factor of n

    how can I show that, any help will be apprciated.

    Plain substitution: n + d^2 = pq+\left(\frac{q-p}{2}\right)^2=<br />
pq+\frac{q^2}{4}-\frac{qp}{2}+\frac{p^2}{4}

    Well, now just check the above is a perfect square (some junior high school algebra is required here)

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2009
    Posts
    35
    Quote Originally Posted by tonio View Post
    Plain substitution: n + d^2 = pq+\left(\frac{q-p}{2}\right)^2=<br />
pq+\frac{q^2}{4}-\frac{qp}{2}+\frac{p^2}{4}

    Well, now just check the above is a perfect square (some junior high school algebra is required here)

    Tonio
    thanks, but how about saying that square (p+q)^2/2 is n+d^2 so n+d^2 is perfect sequare. also, how can we find the prime factor of n
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jan 2009
    Posts
    591
    Quote Originally Posted by koko2009 View Post
    Let n=pq and q-p=2d where d>0 and p, q are odd prime numbers.



    1-Show that n+d2 is a perfect square i.e. n+d2=m2 for some integer m.
    2-If you know that n+d2 is a perfect square, can you find prime factor of n

    how can I show that, any help will be apprciated.
    (2) n+d^2 = m^2

    n = m^2-d^2

    n = (m+d)(m-d)

    q=(m+d) & p=(m-d)

    or: p=(m+d) & q=(m-d)

    .

    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by koko2009 View Post
    thanks, but how about saying that square (p+q)^2/2 is n+d^2 so n+d^2 is perfect sequare. also, how can we find the prime factor of n
    "How about saying that..."? You have to prove it, of course. And abou that odd question about THE odd prime factor of n: n = qp, so both q, p are odd factors of n...

    Tonio
    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: 13
    Last Post: April 10th 2010, 12:57 PM
  4. N! is never a perfect square
    Posted in the Number Theory Forum
    Replies: 7
    Last Post: December 9th 2009, 09:41 AM
  5. Perfect square
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: October 19th 2008, 08:05 PM

Search Tags


/mathhelpforum @mathhelpforum