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As you know, Primorial ($\#$) notion is defined as the product of first $n$ prime numbers. That is, $$ n\# = \prod_{i=1}^nP_i $$

An there is some effect named Primorial Influence (explained here)which prevents the numbers near to a Primorial to be prime. That is,except than $n\# + 1$ or $n\# - 1$ which are Primorial prime for sum $n$, $n\# + c$ can be prime only if $c \ge P_{n+1}$. As an example the below near Primorial numbers are prime:

$$ P_{1000}\# + P_{1087} \implies \text{3393 digits, }P_{1087} = 8719 \\ P_{1001}\# + P_{1100} \implies \text{3397 digits, }P_{1100} = 8831 \\ P_{1002}\# + P_{1068} \implies \text{3401 digits, }P_{1068} = 8573 $$

From samples above it can be seen that difference of prime index of $n$ and $c$ is less than $100$. I know it can't be generalized to any primorial, but isn't there any good estimation on this upper bound?

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$2\# = 6$ and 6+1 is prime below $6+3$, isn't this a counter-example? – user51427 Dec 9 '12 at 22:33
@sunflower, you're true, I forgot to add $n\# + 1$ and $n\# - 1$ special cases. Question updated. – Mohsen Afshin Dec 10 '12 at 6:19
There probably isn't any good estimate (depending on your definition of good, of course). We can't even prove there's always a prime between $n$ and $n+\sqrt n$. – Gerry Myerson Dec 10 '12 at 6:30
The prime number theorem implies $n\#\approx e^{p_n}$, so Cramer's conjecture implies the largest gaps are $<p_n^2$. Conceivably this supposed "influence" may cause these eventually to be among the largest of the gaps (speculation), but primorials are not the only such numbers; Factorials and highly composite/highly abundant numbers could also be called "influential," (but perhaps less so) the common feature being a large portion of small factors, a notion effectively captured by practical numbers, of which all these but highly abundants (known exceptions are $3,10$) are known to be subsets. – Jaycob Coleman Nov 8 '13 at 3:43

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