I'm having a hard time trying to solve the following problem:
Given any random even natural number, $x$, prove that it can or cannot be written as the product of some integer, $b$, times the primorial function with an argument being some other integer, $n$, s.t. $1< n \leq a$, where $a$ is the smallest integer s.t. $p_a\#\geq x$ and $b$ can be expressed as the sum of coprime partitions of other integers, but each partition must be coprime to $p_n\#$.
For example: if the natural number is $2^{448}-1$, can it be written as $b*p_{n}\#$ or not?
I've tried transforming the problem into another problem where instead I have multiple "coefficients" like $b$ and different arguments like $n$, then summing them. Sort of like a "primorial decomposition".
Perhaps this is relevant? Showing $\prod\limits_{p \leq x} p> e^{(1+\epsilon )x}$ and $\prod\limits_{p \leq x} p < e^{(1-\epsilon) x}$ are false for $x$ large enough.