Primorial — Definition, Formula & Examples

Primorial is the product of all prime numbers less than or equal to a given prime

p

. It is written

p\#

, read as "p primorial."

For a prime number $p$ , the primorial $p\#$ is defined as $p\# = \displaystyle\prod_{\substack{q \leq p \\ q \text{ prime}}} q$ . That is, it equals the product of every prime from $2$ up to and including $p$ .

Key Formula

p\# = 2 \times 3 \times 5 \times 7 \times \cdots \times p

Where:

$p$ = A prime number that serves as the upper bound
$p\#$ = The product of all primes up to and including p

Worked Example

Problem: Find the primorial 11#.

Step 1: List all primes less than or equal to 11.

2,\; 3,\; 5,\; 7,\; 11

Step 2: Multiply them together.

2 \times 3 = 6,\quad 6 \times 5 = 30,\quad 30 \times 7 = 210,\quad 210 \times 11 = 2310

Answer:

11\# = 2310

Why It Matters

Primorials appear in proofs about the distribution of prime numbers, including Euclid's classic proof that infinitely many primes exist (consider

p\# + 1

). They also show up in number theory competitions when constructing numbers with specific divisibility properties.

Common Mistakes

Mistake: Confusing primorial with factorial by multiplying all integers up to p instead of only the primes.

Correction: A factorial uses every positive integer (

5! = 120

), while a primorial uses only primes (

5\# = 2 \times 3 \times 5 = 30

). Always filter for primes first.

Related Terms

Factorial — Product of all positive integers up to n
Double Factorial — Product skipping every other integer
Multifactorial — Generalized factorial with step size k
Subfactorial — Counts derangements, another factorial variant