$T_1=x+1/x$ is an integer greater than $2$. Prove that $x^n+x^{-n}$ is an integer. For what values on $n$, $t_1$ divides $t_n$.
I am stuck with this problem. Please help.
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You can find a few posts related to the first part of the problem using Approach0. For example: $r+r^{-1}$ integral implies $r^n+r^{-n}$ integral or $S_n$ is an integer for all integers $n$. More tips on searching: How to search on this site? – Martin Sleziak Jan 19 '19 at 12:24
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See https://math.stackexchange.com/questions/936479/proving-that-frac-phi4001-phi200-is-an-integer – lab bhattacharjee Jan 19 '19 at 14:07
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$(x^{n+1}+\frac 1 {x^{n+1}})=(x^{n}+\frac 1 {x^{n}}) (x+\frac 1 x)-(x^{n-1}+\frac 1 {x^{n-1}})$. Hence $T_{n+1}=T_nT_1-T_{n-1}$. Thus $T_1$ divides $T_{n-1}$ iff it divides $T_{n+1}$. Can you answer the second part using this?
Kavi Rama Murthy
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oh thank you so much.i'd grateful if you consider answering the second part too,i am not sure about it – Super MaxLv4 Jan 19 '19 at 14:16