Primorial

For n ≥ 2, the primorial (n#) is the product of all prime numbers less than or equal to n. For example, 210 is a primorial which is the product of the first four primes multiplied together (2·3·5·7). The name is attributed to Harvey Dubner. The first few primorials are

2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410. Template:OEIS

They grow rapidly.

The idea of multiplying all primes occurs in a proof of the infinitude of the prime numbers; it is applied to show a contradiction in the idea that the primes could be finite in number.

Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, 2236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with 5136341251. 23# is also the common difference in arithmetic progressions of fifteen and sixteen primes.

Every highly composite number is a product of primorials (e.g. 360 = 2·6·30).

Table of primorials

pp#
22
36
530
7210
112310
1330030
17510510
199699690
23223092870
296469693230
31200560490130
377420738134810
41304250263527210
4313082761331670030
47614889782588491410
5332589158477190044730
591922760350154212639070
61117288381359406970983270
677858321551080267055879090
71557940830126698960967415390
7340729680599249024150621323470
793217644767340672907899084554130
83267064515689275851355624017992790
8923768741896345550770650537601358310
972305567963945518424753102147331756070

References

• Factorial and primorial primes. J. Recr. Math., 19, 1987, 197-203it:Primoriale
Navigation

• Art and Cultures
• Art (https://academickids.com/encyclopedia/index.php/Art)
• Architecture (https://academickids.com/encyclopedia/index.php/Architecture)
• Cultures (https://www.academickids.com/encyclopedia/index.php/Cultures)
• Music (https://www.academickids.com/encyclopedia/index.php/Music)
• Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
• Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
• Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
• Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
• Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
• Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
• History (http://www.academickids.com/encyclopedia/index.php/History)
• Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
• Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
• Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
• Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
• Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
• Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
• United States (http://www.academickids.com/encyclopedia/index.php/United_States)
• Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
• World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
• Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
• Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
• Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
• Science (http://www.academickids.com/encyclopedia/index.php/Science)
• Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
• Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
• Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
• Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
• Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
• Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
• Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
• Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
• Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
• Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
• Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
• Government (http://www.academickids.com/encyclopedia/index.php/Government)
• Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
• Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
• Space and Astronomy
• Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
• Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
• Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
• Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
• Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
• US States (http://www.academickids.com/encyclopedia/index.php/US_States)

Information

• Home Page (http://academickids.com/encyclopedia/index.php)
• Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

• Clip Art (http://classroomclipart.com)