We consider equations of the form [Formula: see text], where [Formula: see text] is a polynomial with integral coefficients and [Formula: see text] is the [Formula: see text]th Fibonacci number that is, [Formula: see text] and [Formula: see text] for [Formula: see text] In particular, for each [Formula: see text], we prove the existence of a polynomial [Formula: see text] of degree [Formula: see text] such that the Diophantine equation [Formula: see text] has infinitely many solutions in positive integers [Formula: see text]. Moreover, we present results of our numerical search concerning the existence of even degree polynomials representing many Fibonacci numbers. We also determine all integral solutions [Formula: see text] of the Diophantine equations [Formula: see text] for [Formula: see text] and [Formula: see text].

中文翻译：

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丢番图方程 Fn = P(x)

我们考虑 [Formula: see text] 形式的方程，其中 [Formula: see text] 是具有整数系数的多项式，[Formula: see text] 是 [Formula: see text] 斐波那契数，即 [Formula: see text] 和 [Formula: see text] for [Formula: see text] 特别是，对于每个 [Formula: see text]，我们证明了 [Formula: see text] 次多项式 [Formula: see text] 的存在性这样丢番图方程 [公式：见正文] 有无限多个正整数解 [公式：见正文]。此外，我们提出了关于表示许多斐波那契数的偶次多项式的存在的数值搜索结果。我们还为 [公式：见文本] 和 [公式：见文本] 确定了丢番图方程 [公式：见文本] 的所有积分解 [公式：见文本]。