There are an infinite number of primes $p$ such that there are no prime $q<p$ such that $p-q$ is $2$-smooth (a power of two) and the smallest such numbers are $p=2,127,149,...$. https://oeis.org/A065381.
The same idea for $3$-smooth differences yields the question if there are prime numbers $p>2$ such that it doesn't exists a prime $q<p$ such that $p-q$ is $3$-smooth? Due to my tests such a prime is greater than $10^{10}$ if it exists.
I do think there is such a failing prime but that my computation skills and computer systems are too limited to find out.
Question: is there a failing prime or can it be proved that $3$-smooth differences always exists?
Comment shows there should be a lot of failing primes, but which is the smallest greater than $2$?
If $F=\{k\in\mathbb N_{>1} |2^{k-1}\equiv 1\pmod k\}$ one can ask for which $n\in F$ it doesn't exist a $m\in F,\; m<n$ such that $n-m$ is $3$-smooth. For all $n\in \{t\in F | 10^{10}\leq t < 2\cdot10^{10}\}$ there exist one such number $19115108601$, which is not a prime.
A big gap from $p$ downwards to previous prime should decrease the probabillity for 3-smooth gaps. I tested all so far known records of gaps
without find an exception. So I add the conjecture tag.
Below a diagram of the number of 3-smooth differences as a function of the prime $p$. The red curve is $\ln p$
