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Math Help - Difficulties proving that a quadratic expression is composite

  1. #1
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    Difficulties proving that a quadratic expression is composite

    I am trying to prove the following statement:
    There is a quadratic f(n) = n2 + bn + c with positive coefficients b and c, such that f(n) is composite.
    I am facing difficulties on how to approach this proof and right now I have no idea on how to start. Could you please give me a hint?

    Thank you very much in advance.
    Last edited by matemauch; October 19th 2012 at 12:33 PM.
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  2. #2
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    Re: Difficulties proving that a quadratic expression is composite

    Quote Originally Posted by matemauch View Post
    I am trying to prove the following statement:
    There is a quadratic f(n) = n2 + bn + c with positive coefficients a and b, such that f(n) is composite.
    Where is any a?
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  3. #3
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    Re: Difficulties proving that a quadratic expression is composite

    I am sorry for the typo. It should be "...with positive integers b and c, such that..."

    Thanks !
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    Re: Difficulties proving that a quadratic expression is composite

    Quote Originally Posted by Plato View Post
    Where is any a?
    Sorry for the typo. I think now it is correct.
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  5. #5
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    Re: Difficulties proving that a quadratic expression is composite

    Take b= 3, c= 2. Then n^2+ 3n+ 2= (n+1)(n+2) is, for any n, the product of two integers and so composite.
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  6. #6
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    Re: Difficulties proving that a quadratic expression is composite

    Quote Originally Posted by HallsofIvy View Post
    Take b= 3, c= 2. Then n^2+ 3n+ 2= (n+1)(n+2) is, for any n, the product of two integers and so composite.
    Thank you very much. I completely forgot about the factorization. Based on your suggestion I elaborated the next line of thought:
    Let q and p positive integers.
    Define b = x+y, and b is a positive integer
    Define c = xy, and c is a positive integer
    Then n^2 + bn + c = n^2 + (x+y)n + xy = (n+x)(n+y) is composite.

    What do you think of it?
    Thank you for your help and your comment.
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  7. #7
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    Re: Difficulties proving that a quadratic expression is composite

    What are you trying to prove here? I am lost in this, from what you are saying, x and y need not even be rational numbers, ok? Is that fine, having some irrational numbers in the factorization?

    Salahuddin
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    Re: Difficulties proving that a quadratic expression is composite

    Quote Originally Posted by Salahuddin559 View Post
    What are you trying to prove here? I am lost in this, from what you are saying, x and y need not even be rational numbers, ok? Is that fine, having some irrational numbers in the factorization?

    Salahuddin
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    Assume x and y be integers
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  9. #9
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    Re: Difficulties proving that a quadratic expression is composite

    It is not always possible that x and y are integers. x and y are such that -x and -y are roots of this quadratic. And those are not always real numbers, even.

    Salahuddin
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