Primorielle

For $n > 1$ , the primorielle N , noted $n \ #$ or $P (N)$ , is the product of all the prime numbers lower (or equal) to N . For example, 210 is primorielle which is the product of the first four prime numbers (2 X 3 X 5 X 7). These numbers were named thus by Harvey Dubner. The first primorielles ones are

2, 6, 30, 210, 2310,30030,510510,9699690,223092870,6469693230,200560490130,7420738134810,304250263527210,13082761331670030,614889782588491410. (see)

The idea to multiply prime numbers consecutive appears in the demonstration of the infinitude of the prime numbers; it is used to show that one can always find a prime number other that those, of finished number, that one knows already.

The primorielles ones play a big role in the research of the prime numbers in additive arithmetic progressions. For example 2236133941 + P (23) gives a prime number, beginning a succession of thirteen prime numbers obtained by adding in a repetitive way P (23), and while finishing by 5136341251+ P (23).
P (23) and is also the common difference between the arithmetic progressions of fifteen and sixteen prime numbers.

All highly made up Nombre is a product of primorielles (example 360 = 2 X 6 X 30).

Count of the primorielles ones

p : P ( p ) ( p first) --- ------------ 2:2 3:6 5:30 7:210 11:2310 13:30030 17:510510 19:9699690 23:223092870 29:6469693230 31:200560490130 37:7420738134810 41:304250263527210 43:13082761331670030 47:614889782588491410 53:32589158477190044730 59:1922760350154212639070 61:117288381359406970983270 67:7858321551080267055879090 71:557940830126698960967415390 73:40729680599249024150621323470 79:3217644767340672907899084554130 83:267064515689275851355624017992790 89:23768741896345550770650537601358310 97:2305567963945518424753102147331756070

See too

Prime number primoriel

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