A primorial, denoted $p_n\#$, is the product of the first $n$ prime numbers ($p_1=2,\ p_2=3$ etc.). The magnitude of primorials grows rapidly beyond the range of convenient arithmetic manipulation. The number $(p_n\#+1)$ is not divisible by any of the first $n$ primes, and so is frequently a prime number itself. For $n=1,2,3,4,5,11$, $(p_n\#+1) \in \mathbb P$.
I noticed (for primorials accessible to calculation) that when $(p_n\#+1) \not \in \mathbb P$, that a 'near primorial' number plus $1$ could be identified that was a prime. By near primorial number, I mean the product of all but one of the first $n$ primes, or $\frac{p_n\#}{p_i};\ 1<i<n$. For example, $\frac{p_8\#}{3}+1,\ \frac{p_{10}\#}{3}+1,\ \frac{p_6\#}{5}+1,\ \frac{p_7\#}{5}+1,\ \frac{p_{12}\#}{7}+1,\ \frac{p_{13}\#}{11}+1,\ \frac{p_{9}\#}{13}+1$ are all primes. Examples of this kind can be rewritten in the form $p_n\#=p_i(p_k-1);\ 1<i<n,\ k>n$.
Based on this admittedly extremely small set, I conjecture that it might be the case $$p_n\#=C(p_k-1);\ C\in \{1,p_i\},\ 1<i<n,\ k>n$$ The signal feature of $C$ is that it is not composite. A single counterexample arrived at by computation would disprove the conjecture, but for $p_{14}\#$ and greater, the numbers are beyond my ability to conveniently calculate.
My questions are: Has this conjecture been previously considered and settled? If not, is there an analytic approach to prove or disprove the conjecture?
Best Answer
A few lines of Mathematica shows that $p_{19}\#$ is the first counterexample. $$p_{19}\# = \bigg(\prod_{i=1}^{19}p_i\bigg)+1=7858321551080267055879091=54730729297\cdot 143581524529603,$$ so it is composite. The following table shows that $\frac{p_{19}\#}{p_n}+1$ is composite for all $n$ satisfying $1\leq n < 19$
$$\begin{array}{|c|c|c|c|} n & p_n & \frac{p_{19}\#}{p_n}+1 & \text{smallest divisor of }\frac{p_{19}\#}{p_n}+1\\ \hline 1 & 2 & 3929160775540133527939546 & 2 \\ \hline 2 & 3 & 2619440517026755685293031 & 613 \\ \hline 3 & 5 & 1571664310216053411175819 & 5501 \\ \hline 4 & 7 & 1122617364440038150839871 & 21713 \\ \hline 5 & 11 & 714392868280024277807191 & 389 \\ \hline 6 & 13 & 604486273160020542759931 & 131 \\ \hline 7 & 17 & 462254208887074532698771 & 101 \\ \hline 8 & 19 & 413595871109487739783111 & 136483 \\ \hline 9 & 23 & 341666154394794219820831 & 26801 \\ \hline 10 & 29 & 270976605209664381237211 & 809 \\ \hline 11 & 31 & 253494243583234421157391 & 127 \\ \hline 12 & 37 & 212387068948115325834571 & 3449 \\ \hline 13 & 41 & 191666379294640659899491 & 3593 \\ \hline 14 & 43 & 182751663978610861764631 & 167 \\ \hline 15 & 47 & 167198330874048235231471 & 71 \\ \hline 16 & 53 & 148270217944910699167531 & 2866463 \\ \hline 17 & 59 & 133191890696275712811511 & 283 \\ \hline 18 & 61 & 128824943460332246817691 & 179 \\ \hline \end{array}$$