A polynomial variant of diophantine triples in linear recurrences
Tóm tắt
Let
$$ (G_n)_{n=0}^{\infty } $$
be a polynomial power sum, i.e. a simple linear recurrence sequence of complex polynomials with power sum representation
$$ G_n = f_1\alpha _1^n + \cdots + f_k\alpha _k^n $$
and polynomial characteristic roots
$$ \alpha _1,\ldots ,\alpha _k $$
. For a fixed polynomial p, we consider sets
$$ \left\{ a,b,c \right\} $$
consisting of three non-zero polynomials such that
$$ ab+p, ac+p, bc+p $$
are elements of
$$ (G_n)_{n=0}^{\infty } $$
. We will prove that under a suitable dominant root condition there are only finitely many such sets if neither
$$ f_1 $$
nor
$$ f_1 \alpha _1 $$
is a perfect square.
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