Primorial primes are a special type of prime numbers related to the product of the first few prime numbers, known as primorials. A Primorial, written as n#, is the product of the first n prime numbers. For example:
- The first prime number is 2.
- The first two prime numbers are 2 and 3, so p2# = 2 × 3 = 6.
- The first three prime numbers are 2, 3, and 5, so p3# = 2 × 3 × 5 = 30.
Primorial Prime
A primorial prime is a prime number that you get by adding or subtracting 1 from a primorial. So, if you take a primorial pn#, then:
- pn#+1
- pn#−1
If either of these results is a prime number, it’s called a primorial prime.
Examples of Primorial Primes
For first few values of n, we can find the Primorial primes as follows:
- Primes from First Primorial
- The first prime is 2, so the first primorial is: p1# = 2.
- p1# + 1 = 2 + 1 = 3 (which is prime).
- p1# − 1 = 2 − 1 = 1 (not prime).
- So, 3 is a primorial prime. (first)
- Primes from Second Primorial
- The first two primes are 2 and 3, so the second primorial is: p2# = 2 × 3 = 6
- p2# + 1 = 6 + 1= 7 (which is prime).
- p2# − 1 = 6 − 1 = 5 (which is also prime).
- So, both 5 and 7 are primorial primes. (second and third)
- The first two primes are 2 and 3, so the second primorial is: p2# = 2 × 3 = 6
- Primes from Third Primorial
- The first three primes are 2, 3, and 5 so the third primorial is: p3# = 2 × 3 × 5 = 30.
- p3# + 1 = 30 + 1 = 31 (which is prime).
- p3# - 1 = 30 - 1 = 29 (not prime).
- So, both 31 and 29 are primorial prime. (fourth and fifth)
- The first three primes are 2, 3, and 5 so the third primorial is: p3# = 2 × 3 × 5 = 30.
- Primes from Fourth Primorial:
- The first four primes are 2, 3, 5, and 7, so the fouth primorial is: p4# = 2 × 3 × 5 × 7 = 210
- p4# + 1 = 210 + 1= 211 (which is prime).
- p4# − 1 = 210 − 1 = 209 (not prime).
- So, 211 are primorial prime. (sixth)
First Few Primorial Primes are:
2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309, . . .
Note: Largest known primorial prime (of the form pn# + 1) is 6369619# + 1 with 2765105 digits(As of October 2024).
Conclusion
In conclusion, primorial primes are a unique type of prime number that are either one more or one less than a primorial, which is the product of the first few prime numbers. While there are only a few known primorial primes, they highlight the surprising ways in which prime numbers can appear.
Read More,
