In this post I present a similar question that shows section **A19** of [1], I was inspired in it to define my sequence and question. I am asking about it since I think that the problem that arises from my question harmonizes with this.


For integers $k\geq 1$ we denote the primorial of order $k$ as $N\_k$, that is 


$$N\_k=\prod\_{r=1}^k p\_r$$


where thus $p\_r$ denotes the $r-th$ prime number. I add as reference that Wikipedia has the article [*Primorial.*](https://en.wikipedia.org/wiki/Primorial)


In next paragraph $n\geq 1 $ denotes the integer variable.


**Definition.** *Now we consider the values of* $n$ *making* $n-N\_k$ *prime for all those primorials* $N\_k$ *satisfying* $2\leq N\_k<n$.


I've calculated the first few terms from the definition. Our sequence starts as $$n=4,9,25,43,49,73\ldots...$$


To illustrate our sequence we add the example about why $25$ is in our sequence: the only primorials less than $n=25$ are $N\_1=2$ and $N\_2=6$, and we get that the integers $25-2$ and $25-6$ are both prime numbers.


I have no idea/intutition if this sequence has only a finite number of terms.



> 
> **Question.** I don't know if it exists only a finite number of positive integers $n's$ being $$n-N\_k$$ a prime number for all primorial $2\leq N\_k<n$. What work can be done to study if the sequence of $n's$ from this definition should have a finite number of terms? **Many thanks.**
> 
> 
> 


I am asking if it is possible to deduce a proposition, get some idea about it, or set some conjecture. 


**I emphasize** that I am asking about what work or statements/heuristics we can get for the **Question,** since it is a hard question (see the comments) I think that I should accept an available and useful answer.


References:
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[1] Richard K. Guy, *Unsolved Problems in Number Theory*, Unsolved problems in Intuitive Mathematics Volume I, Second Edition, Springer-Verlag (1994).



On values of $n\geq 1$ satisfying that for all primorial $N_k$ less than $n$ the difference $n-N_k$ is a prime number
Instead of looking at primorials, one can look at so-called *admissible sets*. We say that a set of integers $S = \{h\_1, \cdots, h\_k\}$ is *admissible* if for all primes $p$, $S \pmod{p}$ is not a complete set of residues. Note that for all $p > |S|$, $S \pmod{p}$ is automatically not a full set of residues.


For example, the set $\{1,2\}$ is not admissible since modulo 2 the set represents a complete set of residues, whereas $\{0,2\}$ is admissible.


The conjecture then (originally attributed to Hardy-Littlewood) that for any admissible set $S$, there exists infinitely many integers $n$ such that each of $n + h\_1, n + h\_2, \cdots, n + h\_k$ is prime. In fact the Hardy-Littlewood $k$-tuple conjectures predicts an asymptotic formula for the number of such $n \leq X$.


This conjecture, of course, is totally out of reach. What is known is due to the seminal work of Yitang Zhang, James Maynard and the two polymath projects: that for for $h\_i \geq 0$ for $i = 1, \cdots, k$ and $h\_i < h\_j$ for $i < j$, whenever $h\_k - h\_1 \geq 246$ there exists infinitely many $n$ such that there exists two indices $i < j$ such that $n + h\_i, n + h\_j$ are both prime.


Maynard also proved the following which is out of reach using Zhang's approach: for each positive integer $m$ there exists a number $B(m)$ such that whenever $S$ is an admissible set of size $k$ exceeding $B(m)$, there exists infinitely many $n$ such that $n + h\_i$ is prime for at least $m$ indices $i$. In fact, one can take $B(m) = O(e^{4m}m^3)$.


These results are known as *bounded gaps between primes*. They represent an extremely high achievement in prime number theory in recent times, perhaps the greatest results in a generation.


