Are the ratios of logarithms of prime numbers dense in $\mathbb{R}^+$? Let $\alpha$ and $\beta$ be two positive reals, $\alpha \lt \beta$ (arbitrarily close). Is it true that there always exists an ordered pair of prime numbers, $(p, q)$, such that
$$ \alpha \lt \frac{\log(q)}{\log(p)} \lt \beta $$
?
Context: I'm interested in classifying the natural numbers by their Factorization Patterns (FPs) and their Factorization Patterns of Sequences of Divisors (FPSDs), two kinds of symbolic signatures based on the prime factorization, where the order of the divisors matters; let's just provide self-explanatory examples:
$$\mbox{FP}(350)=\mbox{FP}(2 \times 5^2 \times 7)=pq^2r$$
$$\mbox{FP}(12)=\mbox{FP}(20)=p^2q$$
$$\mbox{FPSD}(12)=[1 \lt p \lt q \lt p^2 \lt pq \lt p^2q]$$
$$\mbox{FPSD}(20)=[1 \lt p \lt p^2 \lt q \lt pq \lt p^2q]$$
The last 3 lines prove it's possible to find natural numbers with the same FP but with distinct FPSDs. Then comes the question: given an FP, how many distinct FPSD's for that FP can there exist? For example, is it worth looking for a $z$, distinct from $12$ and $20$ such that $\mbox{FP}(z)$ is also equal to $p^2q$ but such that $\mbox{FPSD}(z)$ is neither $[1 \lt p \lt q \lt p^2 \lt pq \lt p^2q]$ nor $[1 \lt p \lt p^2 \lt q \lt pq \lt p^2q]$?
My work led me to arrangements of [vectorial] hyperplanes (by taking the logarithms) where the dimension is the number of distinct primes in the prime factorization of $z$. In the case when the dimension is 2, the arrangement is just a drawing of lines passing through $(0,0)$, like this:
Conjecturally, there are as many FPSD for a given FP as there are colored regions. That supposes that every region contains at least one point with coordinates $(\log(p),\log(q))$ for some $(p,q)$ ordered pair of prime numbers, $p \lt q$. Since the number of delimiting lines tends to grow fast when the exponents $m$ and $n$ increase in $\mbox{FP}=p^m q^n$, the regions become thiner and thiner. Hence the asked question, which I was unsuccessful to prove. I'm wondering if it's a conjecture closer to Bertrand's conjecture (proved -- theorem of Chebychev) or to Goldbach's conjecture (still unproved).
N.B.: this topic also deals with a entry in OEIS, that I'm writing, currently in draft status (A355474).
 A: Yes. More strongly, we claim that, for any $\alpha<\beta$, there exists some $N>0$ so that for all $p>N$, there exists a prime $q$ with $\alpha<\frac{\log q}{\log p}<\beta$, i.e. with $p^\alpha<q<p^\beta$.
Let $N=2^{\frac1{\beta-\alpha}}$, so that for all $p>N$
$$p^\beta\geq N^{\beta-\alpha}p^\alpha=2p^\alpha.$$
Then, by Bertrand's postulate, there exists a prime $q$ between $p^\alpha$ and $2p^\alpha$. Such a prime must also lie between $p^\alpha$ and $p^\beta$.
(If you're interested in more general factorization patterns, you can show using essentially the same logic that the set
$$\left\{\left(\frac{\log q_1}{\log p},\frac{\log q_2}{\log p},\dots,\frac{\log q_n}{\log p}\right)\colon p,q_1,\dots,q_n\text{ prime}\right\}$$
is dense in $\mathbb R^n_{>0}$ for any $n$.)
A: Say you are given an increasing sequence
$$1 < x_1 < x_2 < \ldots$$ and you want to know whether the set of ratios
$$\left(\frac{x_l}{x_k}\right)_{k<l}$$
Take the $\log$ and reduce to the question: is the set of differences
$$(y_l - y_k)_{k<l}$$
dense in $[0, \infty)$. ( $y_n\colon = \log x_n$). Now the set of differences is dense if

*

*$y_n\to \infty$


*$\lim (y_{n+1} - y_n) = 0$
Indeed, start with a small positive $y_{m+1} - y_m$ and keep adding small $(y_{n+1}-y_n)$ for $n\ge m+1$ till you hit your desired interval.
Now apply this to your question. It turns out that the set of rations $\left(\frac{p}{q}\right)_{p,q\  \text{prime}}$ is also dense in $[0, \infty)$, perhaps more interesting.
