A basic problem in transcendental number theory is to determine the arithmetic properties of analytic functions of the form $f(z)={\sum }_{k=0}^{\infty }{a}_k{z}^k$ where the coefficients $a_k\in K$ belong to an algebraic number field. In particular, one of the most basic problems is to determine if $f(\alpha )$ is algebraic or transcendental for non-zero algebraic arguments α. For example, if $f(z)$ is a transcendental Mahler function, then under generic conditions $f(\alpha )$ is transcendental for all non-zero algebraic numbers with $|\alpha |<1$. Also, if $f(z)$ is an E-function, then there exist algorithms which completely determine the arithmetic properties of $f^{(n)}(\alpha )$ for non-zero algebraic numbers α. In contrast to these and other algorithmic results, we construct three functions $f(z)$, $g(z)$, and $h(z)$ with computable rational coefficients for which no algorithms exist that determine if $f(n)\in \mathbb{Q}$, $g^{(n)}(1)\in \mathbb{Q}$, or ${\int }_0^{1}h(z)z^{n}dz\in \mathbb{Q}$ for integral $n\ge 0$. Our results are an application of an undecidable variant of the Collatz Problem due to Lehtonen [9].

超越数论的一个基本问题是确定形式的解析函数的算术性质 $F（ž）={\sum }_{ķ=0}^{\infty }{\mathrm{一种}}_{ķ}{ž}^{ķ}$ 系数在哪里 ${\mathrm{一种}}_{ķ}\in ķ$属于代数数域。特别是，最基本的问题之一是确定$F（\alpha ）$对于非零代数参数α，是代数或先验的。例如，如果$F（ž）$ 是先验的马勒函数，然后在一般条件下 $F（\alpha ）$ 对于所有非零代数来说都是超验的 $|\alpha |<\mathrm{1个}$。另外，如果$F（ž）$是一个E函数，那么存在一些可以完全确定$F^{（ñ）}（\alpha ）$对于非零代数α。与这些和其他算法结果相反，我们构造了三个函数$F（ž）$， $G（ž）$和 $H（ž）$ 具有可计算的有理系数，没有可用于确定是否存在算法的算法 $F（ñ）\in \mathbb{问}$， $G^{（ñ）}（\mathrm{1个}）\in \mathbb{问}$， 要么 ${\int }_0^{\mathrm{1个}}H（ž）{ž}^{ñ}dž\in \mathbb{问}$ 用于积分 $ñ\ge 0$。我们的结果是由于Lehtonen [9]导致的Collat​​z问题的一个不确定的变体的应用。