$\newcommand{\l}{\operatorname{lcm}} \newcommand{\la}{\operatorname{lacm}} $
$$\l(\,\overbrace{101,\,103,\,107,\,109}^{\large\text{consecutive primes}}\,) =121{,}330{,}190 > 100{,}000{,}000.$$
$$ \qquad\qquad\qquad\left. \begin{align} 109 \times 50 & = 5450 \\ 101 \times 54 & = 5454 \\ 107 \times 51 & = 5457 \\ 103 \times 53 & =5459 \end{align} \right\} \longleftarrow \text{ These are very close to each other.} $$ So if we say that the "least approximate common multiple" is $$\la(101,\,103,\,107,\,109) = \text{the 5450s, a set of diameter 9,}$$ then we can say that no number in the range
$$\la(101,\,103,\,107,\,109) \pm \Big( \min\{\,101,\,103,\,107,\,109\,\} -9\Big)$$ is divisible by any of $101,\,103,\,107,\,109.$
Therefore other prime numbers than those four exist. Euclid would assure us that $\l(101,\,103,\,107,\,109)+1 = 121{,}330{,}190$ is not divisible by any of those four, and therefore other prime numbers than those four exist. Euclid actually wrote of LCMs of sets of primes rather than of products, but with distinct primes it's the same thing.
Euclid's way of generating infinitely many primes is iterative: \begin{align} & \Big( \text{next finite set of primes} \Big) \\ = {} & \text{set of prime factors of} \left( 1+ \Big( \l(\text{finite set of primes that we already have}) \Big)\right). \end{align} If you start with a finite set (say the empty set, or your favorite other finite set of primes) and follow Euclid's method, I suspect you don't get all primes, and you get into big numbers fast. If we use LACMs instead, might we get thicker sets of small primes before going to big ones?
And some questions that perhaps don't call for as much speculation:
- How do we find LACMs? Is there an efficient way?
- What has been published or is found in folklore about them?
And a somewhat vaguer question:
- How surprised should we be that $\la(101,\,103,\,107,\,109)$ is so small?