• Oct 24th 2008, 12:44 AM
alexmahone
Let $\displaystyle \alpha$ and $\displaystyle \beta$ be the roots of the quadratic equation $\displaystyle x^2 + mx - 1 = 0$, where m is an odd integer. Let $\displaystyle \lambda_n = \alpha^n + \beta^n$, for $\displaystyle n \ge 0$. Prove that for $\displaystyle n \ge 0$,
(a) $\displaystyle \lambda_n$ is an integer; and
(b) gcd $\displaystyle (\lambda_n, \lambda_{n + 1}) = 1$.
• Oct 24th 2008, 08:46 AM
HallsofIvy
Quote:

Originally Posted by alexmahone
Let $\displaystyle \alpha$ and $\displaystyle \beta$ be the roots of the quadratic equation $\displaystyle x^2 + mx - 1 = 0$, where m is an odd integer. Let $\displaystyle \lambda_n = \alpha^n + \beta^n$, for $\displaystyle n \ge 0$. Prove that for $\displaystyle n \ge 0$,
(a) $\displaystyle \lambda_n$ is an integer; and
(b) gcd $\displaystyle (\lambda_n, \lambda_{n + 1}) = 1$.

If $\displaystyle \alpha$ and $\displaystyle \beta$ are solutions to that equation, then $\displaystyle (x-\alpha)(x-\beta)= x^2+ mx-1$. Multiplying out the left side, $\displaystyle x^2- (\alpha+ \beta)x+ \alpha\beta= mx^2+ mx- 1$

That tells you $\displaystyle \alpha+ \beta= -m$ and $\displaystyle \alpha\beta= -1$. Does that help?