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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.

user3108815
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    Editing the problem above, you caught that the number should be even, thanks! – user3108815 May 06 '22 at 19:24
  • Oh wait there's a whole other section to the problem. Hold on, editing above. – user3108815 May 06 '22 at 19:33
  • OK, seems to make sense now. – Peter May 06 '22 at 19:43
  • If we make the primorial-part as big as possible, the representation is unique and every even number can be written in the desired way. – Peter May 06 '22 at 19:45
  • It is not difficult to find out the primorial-part in practice. Just multiply the primes $2,3,5\cdots$ together and stop as soon as the next prime factor does not divide the given number. – Peter May 06 '22 at 19:46
  • But this is still not a "primorial-base-system" since we have not something like "digits" as in the factorial-system. – Peter May 06 '22 at 19:49
  • edited the problem, apparently b must include coprime partitions. – user3108815 May 06 '22 at 20:00
  • Let us continue this discussion in chat. – user3108815 May 06 '22 at 20:48
  • I'm having a hard time understanding your broad question in light of your specific example, $2^{448}=b\cdot p_n#$. If $p_n#$ contains any prime factors other than $2$, it cannot be multiplied by any integer $b$ to give a product that has only $2$ in its prime factorization. I conclude that $n=1$ and $b=2^{447}$. I can't figure out what you are asking about $b$ that would allow it to take on that value. – Keith Backman Jun 01 '22 at 17:54
  • @KeithBackman I used an odd number in the question, I think it was very late for me when I copied this question, I had to make quite few edits. – user3108815 Jun 16 '22 at 00:02

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