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Let $21x_{i+3}+20x_{i+2}+19x_{i+1}=x_i^2$, where $i\in\{1,2,\ldots,n\}$, $x_1,x_2,\ldots,x_n\in\mathbb{R}^+$, and $n\in\mathbb{Z}^+$ and $n>3$ (indices are taken mod $n$). Find all $\{x_i\}$ satisfying this equation.

Taking $\mod{2}$ and $\mod{3}$ tells us a little bit about the nature of the sequences but I haven't been able to get any further.

Integrand
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thor077
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    How can you take mod $2$ and $3$ when the $x_i$ are real numbers? – YiFan Tey Oct 14 '20 at 23:51
  • Is $x_i = 0,$ for all $i$ a valid sequence? Also, (and you are not going to like this), my first approach is to express $x_4, x_5, x_6, x_7,$ and $x_8$ in terms of $x_1, x_2, x_3$, and stretch my intuition, seeing where this leads, looking for a pattern. – user2661923 Oct 15 '20 at 00:52
  • Also $x_i=60$ for all $i$ works for any $n$. In general $\min_ix_i\leq60\leq\max_ix_i$. – Servaes Oct 15 '20 at 00:57
  • @YiFan Admittedly, it didn't reveal too much. – thor077 Oct 15 '20 at 03:45
  • @user2661923 I tried this for a long time, didn't find a pattern :( – thor077 Oct 15 '20 at 03:45
  • @Servaes This is true, and I see why. I'm trying to characterize all of the solutions though. – thor077 Oct 15 '20 at 03:46
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    In general, the whole point of the site is that the person who posts the query should show his work in the query. If after posting, he does more work, he should also show that. Whether his work leads anywhere close to a solution is totally irrelevant. A classic example is a query that I posted, where although all my work was wasted, I showed my work:https://math.stackexchange.com/questions/3843823/revisit-456-handshakes-among-91-people – user2661923 Oct 15 '20 at 03:51
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    I disagree with user2661923, I think enough context is shown (even though I really don't understand what you mean by mod $2$ and $3$). – YiFan Tey Oct 15 '20 at 04:46
  • @YiFan +1 : fair point. So, if I understand you correctly, you agree with the objective standard that context is needed, but for this particular query, disagree with my subjective view that work hasn't really been shown. – user2661923 Oct 15 '20 at 11:43
  • @user2661923 Yes, that's accurate. – YiFan Tey Oct 15 '20 at 12:45

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