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Prime numbers problem

  1. Mar 10, 2009 #1

    I can't get this small contest problem. How do you solve this kind of problem?

    Let p and q be prime numbers such that (p^2+q^2)/(p+q) is an integer.
    Prove p=q.
  2. jcsd
  3. Mar 10, 2009 #2
    I can't figure it out.




    p-n1+n2 n1=n2 q
  4. Mar 10, 2009 #3


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    What if r divides the denominator?
  5. Mar 10, 2009 #4
    What's r?
  6. Mar 10, 2009 #5


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    Some number that happens to divide the denominator.
  7. Mar 10, 2009 #6
    I still don't get it, sorry. Can you please explain a little bit more?
  8. Mar 10, 2009 #7
    I think I have a solution but I won't post it without moderator approval.
  9. Mar 11, 2009 #8
    Okay, here is my hint. What theorem might be helpful to show what integer values of p+q will satisfy the following equation?


    Where m is an integer, p is prime and q is prime.
  10. Mar 11, 2009 #9
    I got it.
    Hint: Use the conjugate rule.

    For solution

    but q^2 is only divisible by 1,q,q^2. (p+q)/2 is obv not equal to 1. if it is equal to q, p=q and if it is equal to q^2, q|p and then p=q since they are prime.
    Last edited: Mar 11, 2009
  11. Mar 13, 2009 #10
    This statement can be more generalised as follows;

    Let p and q be prime numbers then (p^2+q^2)/(p+q) is a prime if and only if p = q
  12. Mar 13, 2009 #11
    the prime is p = q
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