关于x^x=y和x^{x^x}=y的无理数作为有理数的无理幂的个数的不可数性及其解的逼近
Proving the uncountability of the number of irrational powers of irrational numbers evaluated as rationals and solutions approximation for x^x=y and x^{x^x}=y
摘要：
数字通常用于定义更复杂的数字。例如，两个整数用于定义一个有理数，两个实数用于定义一个复数。可以预期，无理数的无理幂是无理数。尽管如此，无理数的无理幂实际上可能是有理数。对于许多代人来说，这是离散数学教科书和大学课程中逻辑矛盾的非建设性证明。80年代以来，一个构造性的证明在口头上流传，如$\sqrt{2}^{log_\sqrt{2} 3}$等于3。2008年，Lord出版了一份书面证明。本文的第一个贡献是证明了存在不可数个这样的无理数对使得一个对另一个的幂是有理数。Marshall和Tan回答了是否存在一个无理数$a$使得$a^a$是有理数的问题。他们证明了给定$I$ = $(\frac{1}{e})^{(\frac{1}{e},\infty)}$，那么$I$中的每一个有理数要么是无理$a$的形式$a^a$，要么是在非常薄的集合$\{1,4,27,256,...,n^n,...\}$中。对于任何真实的$y$，解析求解方程$x^x=y$似乎是一项具有挑战性的任务。据我们所知，没有从给定的PDMATHAMT中找到$x$的工作。我们证明了$ln((ln(y))<x$对于$y>e$和$x<ln(y)$对于$y> e^e$。因此，本文的第二个贡献是估计真正的$x$，使方程$x^x=y$或$x^{x^x}=y$之一成立，对于一个给定的$y$。
英文摘要：
Numbers are often used to define more complicated numbers. For example, two integers are used to define a rational number and two reals are used to define a complex number. It might be expected that an irrational power of an irrational number would be an irrational number. Despite that, it is actually possible for an irrational power of an irrational number to be a rational number. For many generations, this circulated as a non-constructive proof by contradiction in logic for discrete mathematics textbooks and college courses. Since the 80s, a constructive proof circulated orally, such as $\sqrt{2}^{log_\sqrt{2} 3}$ equals to 3. A written proof was published in 2008 by Lord. The first contribution of this paper is to prove that there is an uncountable number of such pairs of irrational numbers such that the power of one to the other is a rational number. Marshall and Tan answered the question of whether there is a single irrational number $a$ such that $a^a$ is rational. They proved that given $I$ = $(\frac{1}{e})^{(\frac{1}{e},\infty)}$, then every rational number in $I$ is either of the form $a^a$ for an irrational $a$ or is in the very thin set $\{1,4,27,256,...,n^n,...\}$. It seems a challenging task to analytically solve the equation $x^x=y$ for any real $y$. To the best of our knowledge, there is no work on finding $x$ from a given $y$. We proved that $ln((ln(y))<x$ for $y>e$ and $x<ln(y)$, for $y> e^e$. Hence, the second contribution of this paper is to estimate the real $x$ such that either one of the equations $x^x=y$ or $x^{x^x}=y$ holds, for a given $y$.
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