Show that $F^2_{n+2} – F^2_{n-2}$ is not a multiple of a Fibonacci number.
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For $F_n$ as n-th Fibonacci number, I tried for a few first numbers $n=2,3,4,5$ the numerical value of $F^2_{n+2} – F^2_{n-2}$. Unlike the previous exercises of the book, when the r.h.s. was another Fibonacci number, in this case I can't find a relation between $24, 63, 165, dots$ with Fibonacci numbers. Any hint? Knowing the r.h.s of $F^2_{n+2} – F^2_{n-2} = ?$ proof will be straightforward but what is the r.h.s of $F^2_{n+2} – F^2_{n-2} = ?$
fibonacci-numbers
asked Nov 19 at 17:58
72D
50916
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For $F_n$ as n-th Fibonacci number, I tried for a few first numbers $n=2,3,4,5$ the numerical value of $F^2_{n+2} – F^2_{n-2}$. Unlike the previous exercises of the book, when the r.h.s. was another Fibonacci number, in this case I can't find a relation between $24, 63, 165, dots$ with Fibonacci numbers. Any hint? Knowing the r.h.s of $F^2_{n+2} – F^2_{n-2} = ?$ proof will be straightforward but what is the r.h.s of $F^2_{n+2} – F^2_{n-2} = ?$
fibonacci-numbers
asked Nov 19 at 17:58
72D
50916
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
For $F_n$ as n-th Fibonacci number, I tried for a few first numbers $n=2,3,4,5$ the numerical value of $F^2_{n+2} – F^2_{n-2}$. Unlike the previous exercises of the book, when the r.h.s. was another Fibonacci number, in this case I can't find a relation between $24, 63, 165, dots$ with Fibonacci numbers. Any hint? Knowing the r.h.s of $F^2_{n+2} – F^2_{n-2} = ?$ proof will be straightforward but what is the r.h.s of $F^2_{n+2} – F^2_{n-2} = ?$
fibonacci-numbers
asked Nov 19 at 17:58
72D
50916
For $F_n$ as n-th Fibonacci number, I tried for a few first numbers $n=2,3,4,5$ the numerical value of $F^2_{n+2} – F^2_{n-2}$. Unlike the previous exercises of the book, when the r.h.s. was another Fibonacci number, in this case I can't find a relation between $24, 63, 165, dots$ with Fibonacci numbers. Any hint? Knowing the r.h.s of $F^2_{n+2} – F^2_{n-2} = ?$ proof will be straightforward but what is the r.h.s of $F^2_{n+2} – F^2_{n-2} = ?$
fibonacci-numbers
fibonacci-numbers
asked Nov 19 at 17:58
72D
50916
asked Nov 19 at 17:58
72D
50916
edited Nov 20 at 23:26
asked Nov 19 at 17:58
72D
50916
asked Nov 19 at 17:58
72D
50916
asked Nov 19 at 17:58
72D
50916
50916
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Maybe this helps:$$F_{n+2}^2-F_{n-2}^2=(F_{n+2}-F_{n-2})(F_{n+2}+F_{n-2})$$
$$ F_{n+2}-F_{n-2}=F_{n+1}+F_n-F_{n-2}=F_{n+1}+F_{n-1}=F_n+2F_{n-1}$$
$$ F_{n+2}+F_{n-2}=F_{n+1}+F_n+F_{n-2}=2F_n+F_{n-1}+F_{n-2}=3F_n$$
answered Nov 19 at 18:06
Hagen von Eitzen
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Maybe this helps:$$F_{n+2}^2-F_{n-2}^2=(F_{n+2}-F_{n-2})(F_{n+2}+F_{n-2})$$
$$ F_{n+2}-F_{n-2}=F_{n+1}+F_n-F_{n-2}=F_{n+1}+F_{n-1}=F_n+2F_{n-1}$$
$$ F_{n+2}+F_{n-2}=F_{n+1}+F_n+F_{n-2}=2F_n+F_{n-1}+F_{n-2}=3F_n$$
answered Nov 19 at 18:06
Hagen von Eitzen
274k21266494
add a comment |
up vote
1
down vote
accepted
Maybe this helps:$$F_{n+2}^2-F_{n-2}^2=(F_{n+2}-F_{n-2})(F_{n+2}+F_{n-2})$$
$$ F_{n+2}-F_{n-2}=F_{n+1}+F_n-F_{n-2}=F_{n+1}+F_{n-1}=F_n+2F_{n-1}$$
$$ F_{n+2}+F_{n-2}=F_{n+1}+F_n+F_{n-2}=2F_n+F_{n-1}+F_{n-2}=3F_n$$
answered Nov 19 at 18:06
Hagen von Eitzen
274k21266494
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Maybe this helps:$$F_{n+2}^2-F_{n-2}^2=(F_{n+2}-F_{n-2})(F_{n+2}+F_{n-2})$$
$$ F_{n+2}-F_{n-2}=F_{n+1}+F_n-F_{n-2}=F_{n+1}+F_{n-1}=F_n+2F_{n-1}$$
$$ F_{n+2}+F_{n-2}=F_{n+1}+F_n+F_{n-2}=2F_n+F_{n-1}+F_{n-2}=3F_n$$
answered Nov 19 at 18:06
Hagen von Eitzen
274k21266494
Maybe this helps:$$F_{n+2}^2-F_{n-2}^2=(F_{n+2}-F_{n-2})(F_{n+2}+F_{n-2})$$
$$ F_{n+2}-F_{n-2}=F_{n+1}+F_n-F_{n-2}=F_{n+1}+F_{n-1}=F_n+2F_{n-1}$$
$$ F_{n+2}+F_{n-2}=F_{n+1}+F_n+F_{n-2}=2F_n+F_{n-1}+F_{n-2}=3F_n$$
answered Nov 19 at 18:06
Hagen von Eitzen
274k21266494
answered Nov 19 at 18:06
Hagen von Eitzen
274k21266494
answered Nov 19 at 18:06
Hagen von Eitzen
274k21266494
answered Nov 19 at 18:06
Hagen von Eitzen
274k21266494
274k21266494
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