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I have already posted this question at MSE here, but as it received a few upvotes, but no comments or answers I choose to cross-post it here.

Let $P$ be a degree-two polynomial, with roots $\alpha,\beta$. Is there a simple condition on $P$ (or on $\alpha,\beta$), equivalent to the following :

$$ (*)\alpha^i\beta^j-\alpha^j\beta^i+\beta^i-\beta^j+\alpha^j-\alpha^i= \left|\begin{array}{cc} \alpha^i-1 &\beta^i-1 \\ \alpha^j-1 & \beta^j-1 \end{array}\right| \neq 0, \ \text{for any } i,j\in{\mathbb Z}\setminus \lbrace 0\rbrace,i \neq j. $$

If the coefficients of $P$ are integers, is (*) a decidable question ?

Motivation : (*) is equivalent to the fact that any linear recurrence sequence of the form $c_1\alpha^i+c_2\beta^i$ does not attain any value more than twice, unless it is zero.

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  • $\begingroup$ It seems a sufficient condition is that there exists an odd prime $p$ such that $v_p(a)\ne v_p(b)$ and $2v_p(b)<v_p(ac)$, where $P(x)=ax^2+bx+c$. The point is that if $v$ is a valuation on the quadratic field $\mathbb Q(\alpha)=\mathbb Q(\beta)$ extending $v_p$, then (using $\alpha+\beta=-b/a$ and $\alpha\beta=c/a$) $0\ne v(\alpha)\ne v(\beta)\ne0$, which implies that one of the terms in $(*)$ has valuation strictly less than the other five, so their sum is nonzero. $\endgroup$
    – Emil Jeřábek
    Commented Mar 15, 2014 at 19:12
  • $\begingroup$ Sorry, the condition should also include $v_p(c)\ne v_p(b)$. On the other hand, $p=2$ should work too. $\endgroup$
    – Emil Jeřábek
    Commented Mar 15, 2014 at 19:30
  • $\begingroup$ I believe that the multiplicity of order-2 recurrences is well-understood (although I regret not being able to nominate a reference). $\endgroup$
    – Gerry Myerson
    Commented Mar 16, 2014 at 9:40

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