Primorial prime number

In number theory, one is primorial prime (from the English primorial prime ) is a prime number of the mold , wherein the Primfakultät (or primorial ) of (that is the product of the first prime numbers). ${\displaystyle p}$ ${\displaystyle p=p_{n}\#\pm 1}$ ${\displaystyle p_{n}\#}$ ${\displaystyle p_{n}}$ ${\displaystyle n}$

Prime numbers of the form are also called sorrow prime numbers . Prime numbers of the form are also called Euclidean prime numbers . ${\displaystyle p=p_{n}\#-1}$ ${\displaystyle p=p_{n}\#+1}$

Examples

• Be . It is , thus is the product of the first 7 prime numbers, i.e. all prime numbers up to and including . One receives . Thus is not a prime number and therefore also not a primorial prime number. ${\displaystyle p:=p_{7}\#-1}$ ${\displaystyle p_{7}=17}$ ${\displaystyle p_{7}\#=17\#}$ ${\displaystyle n=17}$ ${\displaystyle p_{7}\#=p_{1}\cdot p_{2}\cdot \ldots \cdot p_{7}=2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17=510510}$ ${\displaystyle p=p_{7}\#-1=510510-1=510509=61\cdot 8369\not \in \mathbb {P} }$
• Be . It is , thus is the product of the first 5 prime numbers, i.e. all prime numbers up to and including . One receives . Thus is a prime number and thus also a primorial prime number. ${\displaystyle p:=p_{5}\#+1}$ ${\displaystyle p_{5}=11}$ ${\displaystyle p_{5}\#=11\#}$ ${\displaystyle n=11}$ ${\displaystyle p_{5}\#=p_{1}\cdot p_{2}\cdot \ldots \cdot p_{5}=2\cdot 3\cdot 5\cdot 7\cdot 11=2310}$ ${\displaystyle p=p_{5}\#+1=2310+1=2311\in \mathbb {P} }$
• Be . It is , thus is the "product of the first prime", so . Thus is not a prime number and therefore also not a primorial prime number. ${\displaystyle p:=p_{1}\#-1}$ ${\displaystyle p_{1}=2}$ ${\displaystyle p_{1}\#=2\#}$ ${\displaystyle p_{1}\#=2\#=2}$ ${\displaystyle p=p_{1}\#-1=2-1=1\not \in \mathbb {P} }$
• Be . It is the empty product . Thus is a prime number and thus also a primorial prime number. ${\displaystyle p:=p_{0}\#+1}$ ${\displaystyle p_{0}\#=1}$ ${\displaystyle p=p_{0}\#+1=1+1=2\in \mathbb {P} }$
• For the following one obtains primorial primes of the form : ${\displaystyle n}$ ${\displaystyle p=p_{n}\#-1}$

2, 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, ... ( continuation A057704 in OEIS )

These numbers can also be written in the following form : ${\displaystyle p\#-1}$ ${\displaystyle p}$

3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, ... (sequence A006794 in OEIS )

Example :

In the eighth position of the above two lists is or . This means that the 68th is prime and is prime. ${\displaystyle 68}$ ${\displaystyle 337}$ ${\displaystyle p_{68}=337}$ ${\displaystyle p=p_{68}\#-1=337\#-1\in \mathbb {P} }$

• For the following one obtains primorial primes of the form : ${\displaystyle n}$ ${\displaystyle p=p_{n}\#+1}$

0, 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, ... (sequence A014545 in OEIS )

These numbers can also be written in the following form : ${\displaystyle p\#+1}$ ${\displaystyle p}$

(1), 2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, ... ( Follow A005234 in OEIS )

Example :

In the eighth position of the above two lists is or . This means that the 75th is prime and is prime. ${\displaystyle 75}$ ${\displaystyle 379}$ ${\displaystyle p_{75}=379}$ ${\displaystyle p=p_{75}\#-1=379\#-1\in \mathbb {P} }$

• The following list gives the smallest primorial primes of the form : ${\displaystyle p=p_{n}\#\pm 1}$

2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309, ... (Follow A228486 in OEIS )

• The largest known primorial prime of the form is the following (as of February 12, 2020): ${\displaystyle p=p_{n}\#+1}$

${\displaystyle p=p_{33237}\#+1=392113\#+1}$

It was discovered by Daniel Heuer on September 20, 2001 and has 169,966 positions.

• The largest known primorial prime of the form is the following (as of February 12, 2020): ${\displaystyle p=p_{n}\#-1}$

${\displaystyle p=p_{85586}\#-1=1098133\#-1}$

It was discovered on February 28, 2012 by James P. Burt from the Cayman Islands as part of the PrimeGrid project and has 476,311 jobs.

Unsolved problems

• Are there an infinite number of primorial primes of the form ? ${\displaystyle p=p_{n}\#-1}$
• Are there an infinite number of primorial primes of the form ? ${\displaystyle p=p_{n}\#+1}$

Relation to Euclid's theorem

The Greek mathematician Euclid proved around 300 BC Chr. , That there are infinitely many primes ( Euclid's theorem ). The proof is a proof by contradiction , an assumption is made which turns out to be false in the course of the proof. The assumption must be dropped and the opposite of the assumption must be true:

Assume that there are only finitely many prime numbers . You multiply all these prime numbers together and you get the number . Then the following number must not have a prime divisor that already had, because no number can divide both a number and its successor , except for the number , which is not a prime number (and is also called a unit in mathematics ). But since, according to the assumption, is the product of all existing prime numbers and has none of these prime divisors, it must itself be a (new, previously unknown) prime number, which is in contradiction to the requirement that the only existing prime numbers are. The assumption has to be dropped, so the opposite of the assumption applies, there are infinitely many prime numbers. ${\displaystyle p_{1},p_{2},\ldots ,p_{n}}$ ${\displaystyle m=p_{1}\cdot p_{2}\cdot \ldots \cdot p_{n}}$ ${\displaystyle m+1}$ ${\displaystyle m}$ ${\displaystyle p}$ ${\displaystyle m}$ ${\displaystyle m+1}$ ${\displaystyle n=1}$ ${\displaystyle m}$ ${\displaystyle m+1}$ ${\displaystyle m+1}$ ${\displaystyle p_{1},p_{2},\ldots ,p_{n}}$ ${\displaystyle \Box }$

Now, after studying this proof, one could mistakenly assume that the procedure of multiplying the first prime numbers always leads to new prime numbers. It is not so. It can already be seen from the above examples that the form is only obtained for (primorial) prime numbers . For but not, as you can see in the following example: ${\displaystyle n\in \{0,1,2,3,4,5,11,75,\ldots \}}$ ${\displaystyle p=p_{n}\#+1}$ ${\displaystyle n\in \{6,7,8,9,10,12,13,\ldots ,74,76,77,\ldots \}}$

Let and be the product of the first six prime numbers. So then is . If you add to it you get . In fact, this number is neither divisible by, nor by, or . But the following applies: and therefore is not a prime number. In the rarest of cases, this results in a prime number, as can also be seen from the above examples. ${\displaystyle n:=6}$ ${\displaystyle m:=p_{1}\cdot p_{2}\cdot \ldots \cdot p_{6}}$ ${\displaystyle m=2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13=30030}$ ${\displaystyle 1}$ ${\displaystyle m+1=30031}$ ${\displaystyle 2}$ ${\displaystyle 3,5,7,11}$ ${\displaystyle 13}$ ${\displaystyle 30031=59\cdot 509}$ ${\displaystyle m+1=30031}$

See also

• Euclidean number
• Faculty prime
• Primorial

Web links

• Eric W. Weisstein : Primorial Prime . In: MathWorld (English).
• Chris K. Caldwell: primorial prime. Prime Pages, accessed February 12, 2020 . 
• Harvey Dubner: Factorial and primorial primes. Journal of Recreational Mathematics 19 (3), 1987, accessed February 12, 2020 . 
• PrimeFan: table of the first 100 primorials. 2013, accessed February 12, 2020 . 

Individual evidence

1. ↑ ^{a b} Comments on OEIS A228486
2. ↑ 392113 # + 1 on Prime Pages
3. ↑ ^{a b} Chris K. Caldwell: The Top Twenty: Primorial. Prime Pages, accessed February 12, 2020 . 
4. ↑ 1098133 # - 1 on Prime Pages
5. ↑ 1098133 # - 1 on primegrid.com (PDF)
6. ↑ Message 51000. PrimeGrid Forum, March 2, 2012, accessed on February 12, 2020 . 
7. ↑ Michael Hardy, Catherine Woodgold: Prime Simplicity. The Mathematical Intelligencer 31 (4), September 18, 2009, accessed February 12, 2020 .