Hi Michael,

One way to implement primorial.

primorial[n_?Positive] := Product[Prime[i], {i, 1, n}];

primorial[10]
(* 6469693230 *)

Just for kicks-and-giggles, suppressing a bunch of display output to get a final number:

The Product of the first 10,000 Primes renders a 45,337-digit composite number.

n=100,000: 563,921-digits long.

n=1,000,000: 6,722,809-digits long

n=10,000,000: 77,919,922-digits long (that took a minute to compute!)

n=100,000,000: 885,105,237-digits long. (Heh! That took a while... Almost there)

n=200,000,000: ...we'll have to see how long this takes, assuming it can complete it. ;)

I suspect a 1 billion+ digit Product will fall somewhere between the Products of the first 100mil & 200mil primes. But, what do I know? I just play here. ^_^ (If 200mil primes is over 1bn digits, I can probably narrow it down by halving the difference and halving again 'til narrowed down to exactly how many Primes are needed for the Product to cross the 1bn threshold... Might take a while and a few iterations, tho'...)

EDIT: Well, darn, seems like the Mathematica Kernel craps out after about 20 minutes to 1/2 hour on trying to compute on 200,000,000. Sigh ;)

Hi Bill,

Did you look at the primorial function implementation I provided as an answer? Were you looking for something different?

A back-of-the-envelope computation suggests that to get n digits, one requires in the ballpark of LogIntegral[Log[10.]*10^n] primes. For n=10^9, that's around 112,307,000.

Quick check at 10^5. Approximate the count:

In[1002]:= LogIntegral[Log[10.]*10^5]

(* Out[1002]= 20500.5 *)

Check how many digits we get:

In[1004]:= Log[10., Product[Prime[j], {j, 20500}] // N]

(* Out[1004]= 100068. *)