Show that $F^2_{n+2} – F^2_{n-2}$ is not a multiple of a Fibonacci number.

Question

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 $F^2_{n+2} – F^2_{n-2} = ?$

Answer 1

Maybe this helps:$$F_{n+2}^2-F_{n-2}^2=(F_{n+2}-F_{n-2})(F_{n+2}+F_{n-2})$$ $$ F_{n+2}-F_{n-2}=F_{n+1}+F_n-F_{n-2}=F_{n+1}+F_{n-1}=F_n+2F_{n-1}$$ $$ F_{n+2}+F_{n-2}=F_{n+1}+F_n+F_{n-2}=2F_n+F_{n-1}+F_{n-2}=3F_n$$