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Math Help - Roots of a quadratic

  1. #1
    MHF Contributor alexmahone's Avatar
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    Roots of a quadratic

    Let \alpha and \beta be the roots of the quadratic equation x^2 + mx - 1 = 0, where m is an odd integer. Let \lambda_n = \alpha^n + \beta^n, for n \ge 0. Prove that for n \ge 0,
    (a) \lambda_n is an integer; and
    (b) gcd (\lambda_n, \lambda_{n + 1}) = 1.
    Last edited by alexmahone; October 24th 2008 at 07:53 AM.
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  2. #2
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    Quote Originally Posted by alexmahone View Post
    Let \alpha and \beta be the roots of the quadratic equation x^2 + mx - 1 = 0, where m is an odd integer. Let \lambda_n = \alpha^n + \beta^n, for n \ge 0. Prove that for n \ge 0,
    (a) \lambda_n is an integer; and
    (b) gcd (\lambda_n, \lambda_{n + 1}) = 1.
    If \alpha and \beta are solutions to that equation, then (x-\alpha)(x-\beta)= x^2+ mx-1. Multiplying out the left side, x^2- (\alpha+ \beta)x+ \alpha\beta= mx^2+ mx- 1

    That tells you \alpha+ \beta= -m and \alpha\beta= -1. Does that help?
    Last edited by HallsofIvy; October 24th 2008 at 09:18 AM.
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