Solution to Let α, β be roots of the equation x^2− 3x − 1 = 0. For … - Sikademy
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Archangel Macsika

Let α, β be roots of the equation x^2− 3x − 1 = 0. For each nonnegative integer n, let y_n = α^n + β^n . Show that gcd(y_n, y_(n+1)) = 1 for each nonnegative integer n.

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Let's consider the equation x^2-3x-1=0. The discriminant equals to D=9+4=13. So the equation has irrational roots: \alpha=\frac{3+√3}{2}, \beta=\frac{3-√3}{2} . From the Viet theorem \alpha+\beta=3, \alpha\beta=-1. If n=0 we have y_0=\alpha^0+\beta^0=2 . If n=1 we have y_1=\alpha+\beta=3. So gcd(y_0;y_1)=gcd(2;3)=1. In case n=2 we have y_2=(\alpha+\beta)^2-2\alpha\beta=9+2=11, gcd(y_1;y_2)=1 Let's prove that y_n is natural and gcd(y_n, y_{n+1})=1 with the help of the method of mathematical induction. The base of induction is true. Let the proposition is true for all n=1,...,k\in N , and y_1,y_k \in N. gcd(y_{n-1},y_n)=1. If n=k+1 we have y_{n+1}=(\alpha+\beta)y_n+\alpha\beta y_{n-1} . Si we have y_{n+1} \in N. And gcd(y_k, y_{k+1})=gcd(y_k,y_{k-1})=1.

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