Let $f\in \mathbb{Z}[x]$ and $\ell \in \mathbb{N}$. Consider the set of all $(a_0,a_1,\dots ,a_{\ell })\in {\mathbb{N}}^{\ell +1}$ with $a_i<a_{i+1}$ and $f(a_i)|f(a_{i+1})$ for all $0\le i\le \ell -1$. We say that f satisfies the gap principle of order ℓ if $\lim a_{\ell }/a_0=\infty$ as $a_0\to \infty$. We also define the gap order of $f(x)$ to be the smallest positive integer ℓ such that $f(x)$ satisfies the gap principle of order ℓ. If such ℓ does not exist, we say that $f(x)$ does not satisfy the gap principle. In this article, we prove a conjecture by Chan, Choi and Lam that $f(x)$ does not satisfy the gap principle if and only if $f(x)$ is in the form of $f(x)=A{(Bx+C)}^n$ for some $A,B,C\in \mathbb{Z}$. Moreover, we completely determine the gap order of any polynomial. We will show that if $f(x)$ is not in the form of $A{(Bx+C)}^n$, then $f(x)$ has gap order 2 if $f(x)$ is a quadratic polynomial or a power of a quadratic polynomial; and has gap order 1 otherwise.

让 $F\in \mathbb{ž}[X]$ 和 $\ell \in \mathbb{ñ}$。考虑全部$（{\mathrm{一种}}_0，{\mathrm{一种}}_{\mathrm{1个}}，\dots ，{\mathrm{一种}}_{\ell }）\in {\mathbb{ñ}}^{\ell +\mathrm{1个}}$ 与 ${\mathrm{一种}}_{\mathrm{一世}}<{\mathrm{一种}}_{\mathrm{一世}+\mathrm{1个}}$ 和 $F（{\mathrm{一种}}_{\mathrm{一世}}）|F（{\mathrm{一种}}_{\mathrm{一世}+\mathrm{1个}}）$ 对全部 $0\le \mathrm{一世}\le \ell -\mathrm{1个}$。我们说˚F满足订单的差距原则ℓ如果$\mathrm{林}{\mathrm{一种}}_{\ell }/{\mathrm{一种}}_0=\infty$ 如 ${\mathrm{一种}}_0\to \infty$。我们还定义了$F（X）$是最小的正整数ℓ使得$F（X）$为了满足的差距原则ℓ。如果这样ℓ不存在，我们说$F（X）$不满足差距原则。在本文中，我们证明了Chan，Choi和Lam的猜想，$F（X）$ 当且仅当不满足间隔原则 $F（X）$ 形式为 $F（X）=\mathrm{一种}{（乙X+C）}^{ñ}$ 对于一些 $\mathrm{一种}，乙，C\in \mathbb{ž}$。而且，我们完全确定了任何多项式的间隙阶数。我们将证明$F（X）$ 不是以 $\mathrm{一种}{（乙X+C）}^{ñ}$， 然后 $F（X）$ 缺口顺序为2 $F（X）$是二次多项式或二次多项式的幂；并具有缺口顺序1。