I was wondering whether the set of all numbers that can be expressed as a natural number to the power of another natural number has "infinitely wide" gaps or if there is some upper bound between the distance between two neighboring elements. Or, put mathematically:
Let $M_k := \{ n^k | \forall n \in \mathbb{N} \}$. Then, let $M := \bigcup_{i=2}^\infty M_i$. Does there exist an $N \in \mathbb{N}$, such that for all $m_1, m_2 \in M$ (with $m_1 < m_2$ and where there does not exist a $m_3 \in M$ with $m_1 < m_3 < m_2$, in other words, $m_1$ and $m_2$ are "neighbors" in $M$) we have
$$ |m_1-m_2| < N $$
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I don't know how to tackle this question in its original form, but I did come up with an approach, if, instead of being the union of infinitely many $M_k$'s, $M$ is the union of finitely many $M_k$'s. Let $I \subset \mathbb{N}$ be a finite set of indices and with $I = \{a_1, a_2, \cdots, a_m \}$ for some $m \in \mathbb{N}$. (Also order the $a_i$'s, so $a_1 < a_2 < \cdots < a_m$.) Then let $M = \bigcup_{i \in I} M_i$. Assume that there exists some $N$ with the property described above. Then look at the number
$$ S :=N^{a_1\cdot a_2 \cdot \cdots \cdot a_m}$$
We know that $S \in M_{a_1}$, since $S = N^{a_1\cdot a_2 \cdot \cdots \cdot a_m} = (N^{a_2 \cdot \cdots \cdot a_m})^{a_1}$. Also, $(N^{a_2 \cdot \cdots \cdot a_m} + 1)^{a_1} \in M_{a_1}$, and there is no integer power of $a_1$ between them. And since $a_2 > a_1$, $\cdots$, $a_m > a_1$, we know that for all $k \in \{2, \cdots, m \} \subset \mathbb{N}$ that $(N^{(a_1 \cdot a_2 \cdot \cdots \cdot a_m)/a_k} + 1)^{a_k} > (N^{a_2 \cdot \cdots \cdot a_m} + 1)^{a_1}$.
So $N^{a_2 \cdot \cdots \cdot a_m} + 1)^{a_1}$ and $(N^{a_2 \cdot \cdots \cdot a_m})^{a_1}$ are indeed "neighbors". But
$$ \begin{align*} (N^{a_2 \cdot \cdots \cdot a_m} + 1)^{a_1} - (N^{a_2 \cdot \cdots \cdot a_m})^{a_1} &= (N^{a_2 \cdot \cdots \cdot a_m})^{a_1} + a_1\cdot (N^{a_2 \cdot \cdots \cdot a_m})^{a_1-1} + \cdots + 1^{a_1} \\ &\qquad - (N^{a_2 \cdot \cdots \cdot a_m})^{a_1}\\ &> a_1\cdot (N^{a_2 \cdot \cdots \cdot a_m})^{a_1-1} \\ &> N \end{align*} $$
so such an $N$ does not exist.
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However, this argument doesn't work as $M$ becomes the union of infinitely many $M_k$'s, because the product $a_1\cdot \cdots a_m$ is no longer defined. So I think I need a different approach, but I have no idea how to proceed from here...
For $N \in \mathbb N$, the number of powers that are $\le N$ is at most
$$\lfloor N^{1/2}\rfloor + \lfloor N^{1/3}-1\rfloor + \lfloor N^{1/4}-1\rfloor + \lfloor N^{1/5}-1\rfloor + \cdots$$
(where the $-1$'s are to avoid counting $1$ itself more than once). There are at most $\log_2N$ non-zero terms here, and all of them are $ \le N^{1/2}$. So the number of powers $\le N$ is at most $N^{1/2}\log_2 N$. Thus the average gap between powers $\le N$ is at least $\dfrac{N}{N^{1/2}\log_2 N} = \dfrac{N^{1/2}}{\log_2 N}$, which tends to $\infty$ as $N \to \infty$.
Hence the spacing between powers in unbounded.