Primorial

In mathematics, especially in number theory, something called primorial is very interesting. We write it as " p n # " and it works much like the factorial function. But instead of multiplying every whole number, primorial only multiplies the special numbers called prime numbers.

The word "primorial" was created by a person named Harvey Dubner. It combines ideas about primes with the idea behind "factorial," just like how "factorial" connects to factors. This helps mathematicians study numbers in new and fun ways.

Definition for prime numbers

The primorial, written as (p_n#), is a special way to multiply numbers. Instead of multiplying every number like we do in a factorial, we only multiply the first few prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves, like 2, 3, 5, 7, and so on.

pn# as a function of n, plotted logarithmically.

For example, the primorial (p_5#) means we multiply the first 5 prime numbers:
2 × 3 × 5 × 7 × 11 = 2310.

The first few primorials are:
/wiki/1_(number), /wiki/2_(number), /wiki/6_(number), /wiki/30_(number), /wiki/210_(number), /wiki/2310_(number), 30030, 510510, 9699690... (sequence A002110 in the OEIS).

Definition for natural numbers

n ! {\displaystyle n!} (yellow) as a function of ⁠ n {\displaystyle n} ⁠, compared to n # {\displaystyle n\#} (red), both plotted logarithmically.

For any whole number n, its primorial, written as n#, is the product of all prime numbers that are less than or equal to n.

For example, the primorial of 12, written as 12#, is calculated by multiplying all the prime numbers up to 12: 2 × 3 × 5 × 7 × 11 = 2310.

When n is not a prime number, the primorial n# is the same as the primorial of the number before it, (n-1)#.

Properties

The primorial is a special kind of number, much like a factorial, but it only uses prime numbers. Imagine you have a list of prime numbers — these are numbers greater than 1 that can only be divided by 1 and themselves, like 2, 3, 5, 7, and so on. A primorial multiplies these prime numbers together up to a certain point.

For example, if we take the first three prime numbers (2, 3, and 5), their primorial would be 2 × 3 × 5 = 30. This special number has interesting patterns and connections to other areas of mathematics. One cool fact is that the sum of the reciprocals (or one divided by each primorial) gets closer and closer to a specific number, just like adding smaller and smaller pieces together.

Table of primorials

n	n#	p_n	p_n#	Primorial prime?
n	n#	p_n	p_n#	p_n# + 1	p_n# − 1
0	1	—N/a	1	Yes	No
1	1	2	2	Yes	No
2	2	3	6	Yes	Yes
3	6	5	30	Yes	Yes
4	6	7	210	Yes	No
5	30	11	2310	Yes	Yes
6	30	13	30030	No	Yes
7	210	17	510510	No	No
8	210	19	9699690	No	No
9	210	23	223092870	No	No
10	210	29	6469693230	No	No
11	2310	31	200560490130	Yes	No
12	2310	37	7420738134810	No	No
13	30030	41	304250263527210	No	Yes
14	30030	43	13082761331670030	No	No
15	30030	47	614889782588491410	No	No
16	30030	53	32589158477190044730	No	No
17	510510	59	1922760350154212639070	No	No
18	510510	61	117288381359406970983270	No	No
19	9699690	67	7858321551080267055879090	No	No
20	9699690	71	557940830126698960967415390	No	No
21	9699690	73	40729680599249024150621323470	No	No
22	9699690	79	3217644767340672907899084554130	No	No
23	223092870	83	267064515689275851355624017992790	No	No
24	223092870	89	23768741896345550770650537601358310	No	Yes
25	223092870	97	2305567963945518424753102147331756070	No	No
26	223092870	101	232862364358497360900063316880507363070	No	No
27	223092870	103	23984823528925228172706521638692258396210	No	No
28	223092870	107	2566376117594999414479597815340071648394470	No	No
29	6469693230	109	279734996817854936178276161872067809674997230	No	No
30	6469693230	113	31610054640417607788145206291543662493274686990	No	No
31	200560490130	127	4014476939333036189094441199026045136645885247730	No	No
32	200560490130	131	525896479052627740771371797072411912900610967452630	No	No
33	200560490130	137	72047817630210000485677936198920432067383702541010310	No	No
34	200560490130	139	10014646650599190067509233131649940057366334653200433090	No	No
35	200560490130	149	1492182350939279320058875736615841068547583863326864530410	No	No
36	200560490130	151	225319534991831177328890236228992001350685163362356544091910	No	No
37	7420738134810	157	35375166993717494840635767087951744212057570647889977422429870	No	No
38	7420738134810	163	5766152219975951659023630035336134306565384015606066319856068810	No	No
39	7420738134810	167	962947420735983927056946215901134429196419130606213075415963491270	No	No
40	7420738134810	173	166589903787325219380851695350896256250980509594874862046961683989710	No	No

Definition for prime numbers

Definition for natural numbers

Properties

Table of primorials

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