Does the following argument make sense?
Let ${\rm lcm}(1,...,N)$ be the least common multiple of the first $N$ natural numbers. In:
Highest Twin primes such that the number in between twins is the $\text{lcm}$ of first $N$ numbers
it is numerically shown that $N=47$ is the largest $N$ such that ${\rm lcm}(1,...,N) \pm 1$ are twin primes, for $N<36000$.
According to Brun's theorem the number of twin primes less than $n$, $\pi_2(n)$, does not exceed the following expression
$$\pi_2(n) < \frac{Cn}{\log^2n} $$
for some constant $C>0$. The ${\rm lcm}(1,...,N)$ can be written as
$${\rm lcm}(1,...,N)=e^{\psi(N)}$$
where $\psi(N)$ is the Chebyshev function that fulfills
$$|\psi(N)-N| = O\left(N^{\frac{1}{2}+\epsilon}\right)$$
for any $\epsilon > 0$. Imagine we compute the first N ${\rm lcm}(1,...,j)$ with $j\leq N$, such that $n={\rm lcm}(1,...,N)=e^{\psi(N)}$. The probability $P$ that one of the ${\rm lcm}(1,...,j)$ is between 2 twin primes is as maximum given by
$$P < \frac{\frac{Ce^{\psi(N)}}{\log^2e^{\psi(N)}} }{e^{\psi(N)}} = \frac{C}{\psi^2(N)} $$
and hence the number of ${\rm lcm}(1,...,j)$ in between twin primes is maximum $N\times P$,
$$ N\times P < \frac{CN}{\psi^2(N)} $$
which in the limit for very large $N$ vanishes,
$$\lim_{N \to \infty} \frac{CN}{\psi^2(N)} = 0$$
so in the limit $N\times P=0$, which cannot be consistent with infinite ${\rm lcm}(1,...,j)$ between Twin primes, right?
Note that the same argument for the number of single primes next to ${\rm lcm}(1,...,j)$ is consistent with infinite.