论文标题
线性复发中双三元三元的多项式变体
A Polynomial Variant of Diophantine Triples in Linear Recurrences
论文作者
论文摘要
令$（g_n）_ {n = 0}^{\ infty} $为多项式功率总和，即具有功率总和表示复杂多项式的简单线性复发序列$ g_n =f_1α_1α_1^n + \ cdots + cdots + cdots +f_kα_k^n $ and polynomial tartemistal $ rdeistic $ a $ a $。对于固定的多项式$ p $，我们认为成对的不同的非零多项式的三倍$（a，b，c）$，这样$ ab+p，ac+p，bc+p $是$（g_n）_ {n = 0}^{\ infty} $的元素。我们将证明，在合适的主要根本条件下，只有$ f_1 $也不是$f_1α_1$是一个完美的正方形，只有有限的三倍。
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_1α_1^n + \cdots + f_kα_k^n $ and polynomial characteristic roots $ α_1,\ldots,α_k $. For a fixed polynomial $ p $, we consider triples $ (a,b,c) $ of pairwise distinct 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 triples if neither $ f_1 $ nor $ f_1 α_1 $ is a perfect square.
