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From Knuth's The Art of Computer Programming, Volume 1 on page 81 he gives Cassini's identity

$F_{n+1} F_{n-1} - F_n^2 = (-1)^n$. Relation(4)

and follows by saying "Relation(4) shows that $F_n$ and $F_{n+1}$ are relatively prime, since any divisor would have to be a divisor of $(-1)^n$".

I don't see how this result flows from Cassini's identity. Can someone explain this to me?

almel
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    Isn't it trivial? Assume that $d>1$ divides both $F_{n}$ and $F_{n+1}$. Then it divides both $F_{n+1}F_{n-1}$ and $F_n^2$, so it divides $F_{n+1}F_{n-1}-F_n^2$. – Jack D'Aurizio Jun 06 '15 at 13:25

2 Answers2

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Suppose that there is an integer $m\gt 1$ such that $$F_{n}=ms,\ \ \ F_{n+1}=mt$$ where $s,t$ are positive integers. Then from (4) we have that $m$ divides $(-1)^n$, which is impossible.

mathlove
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Let $a,b$ be integers. Every number of the form $c=ax+by$ with $x,y$ integers is a multiple of the GCD of $a$ and $b$. Note if $d \mid a$ and $d \mid b$, then $d\mid ax$ and $d \mid by$ and $d \mid ax +by$.

Now, apply this with $a=F_{n+1}$ and $b=F_n$ and $x= F_{n-1}$ and $y=-F_n$ to get that $(-1)^n$ is a multiple of the GCD of $F_{n+1}$ and $F_n$. So the gcd divides $(-1)^n$ and is thus $1$.

quid
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