Roots of a quadratic

**alexmahone** · October 24th 2008, 12:44 AM

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 $(\lambda_n, \lambda_{n + 1}) = 1$ .

**HallsofIvy** · October 24th 2008, 08:46 AM

Originally Posted by alexmahone

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?

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