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Given a vector of real numbers $a_0, a_1, ... a_n$ where $a_i > 1$ for all $i$, find an $n \in \mathbb{R}$ where $n > 1$ such that $t_i = n^{a_i}$ and $t_i \in \mathbb{N}$ for every $i$. Trivially, $n = 0$ and $n = 1$ will always work but do not apply to my application, and are part of the reason for the $n > 1$ constraint. I am only hopeful that there will be an $n$ for every set of real numbers $a_0, a_1, ... a_n$, but have no good reason to believe that this is necessarily the case. I would be interested in knowing a method to find the smallest number $n$ that meets these criteria, but would be satisfied to find just any.

What general field of mathematics studies problems like this? What methods are typically employed to get an answer? Does every vector admit an a solution $n$?

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Finding the base of exponentiation subject to range constraints.

Given a vector of real numbers $a_0, a_1, ... a_n$ where $a_i > 1$ for all $i$, find an $n \in \mathbb{R}$ where $n > 1$ such that $t_i = n^{a_i}$ and $t_i \in \mathbb{N}$ for every $i$. Trivially, $n = 0$ and $n = 1$ will always work but do not apply to my application, and are part of the reason for the $n > 1$ constraint. I am only hopeful that there will be an $n$ for every set of real numbers $a_0, a_1, ... a_n$, but have no good reason to believe that this is necessarily the case. I would be interested in knowing a method to find the smallest number $n$ that meets these criteria, but would be satisfied to find just any.

What general field of mathematics studies problems like this? What methods are typically employed to get an answer? Does every vector admit an solution $n$?