Near prime

A -fast prime number or also -th order near- prime number is a natural number whose prime factorization consists of exactly prime numbers , with multiple prime divisors being counted accordingly often. Since all natural numbers from prime factors put together are, every natural number is also a fast prime. Near-prime numbers of the second order are also called semi- prime numbers . Near primes move between the poles of the indivisible prime numbers and the maximally divisible highly composite numbers and include both. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$

The Norwegian Viggo Brun introduced the term around 1915 to generalize prime numbers in order to find a new approach to unsolved prime number problems.

definition

Let and with prime numbers . Then it is called the -th order almost prime , where applies. The numbers sequence for a solid is also denoted by. The well-definedness follows from the uniqueness of the prime factorization for all natural numbers. ${\displaystyle z\in \mathbb {N} \setminus \{0\}}$ ${\displaystyle z=\prod _{i=1}^{k}{p_{i}}^{e_{i}}}$ ${\displaystyle p_{1},\dotsc ,p_{k}}$ ${\displaystyle z}$ ${\displaystyle n}$ ${\displaystyle n=\sum _{i=1}^{k}e_{i}}$ ${\displaystyle n}$ ${\displaystyle P_{n}}$

This concept can easily be generalized to the whole numbers and any ZPE rings .

Examples and values

Examples:

• ${\displaystyle 13}$ is a first-order near-prime number ("prime number").
• ${\displaystyle 91=7\cdot 13}$ is a second-order near-prime number ("semi-prime number").
• ${\displaystyle 100=2^{2}\cdot 5^{2}}$ is a fourth order almost prime.
• ${\displaystyle 1024=2^{10}}$ is a tenth order almost prime.
• ${\displaystyle 5308416=2^{16}\cdot 3^{4}}$ is an almost prime twentieth order.

The first twelve first to twentieth order fast primes

01st order	2	3	5	7th	11	13	17th	19th	23	29	31	37	...	Follow A000040 in OEIS
02nd order	4th	6th	9	10	14th	15th	21st	22nd	25th	26th	33	34	...	Follow A001358 in OEIS
03rd order	8th	12	18th	20th	27	28	30th	42	44	45	50	52	...	Follow A014612 in OEIS
04th order	16	24	36	40	54	56	60	81	84	88	90	100	...	Follow A014613 in OEIS
05th order	32	48	72	80	108	112	120	162	168	176	180	200	...	Follow A014614 in OEIS
06th order	64	96	144	160	216	224	240	324	336	352	360	400	...	Follow A046306 in OEIS
07th order	128	192	288	320	432	448	480	648	672	704	720	800	...	Follow A046308 in OEIS
08th order	256	384	576	640	864	896	960	1296	1344	1408	1440	1600	...	Follow A046310 in OEIS
09th order	512	768	1152	1280	1728	1792	1920	2592	2688	2816	2880	3200	...	Follow A046312 in OEIS
10th order	1024	1536	2304	2560	3456	3584	3840	5184	5376	5632	5760	6400	...	Follow A046314 in OEIS
11th order	2048	3072	4608	5120	6912	7168	7680	10368	10752	11264	11520	12800	...	Follow A069272 in OEIS
12th order	4096	6144	9216	10240	13824	14336	15360	20736	21504	22528	23040	25600	...	Follow A069273 in OEIS
13th order	8192	12288	18432	20480	27648	28672	30720	41472	43008	45056	46080	51200	...	Follow A069274 in OEIS
14th order	16384	24576	36864	40960	55296	57344	61440	82944	86016	90112	92160	102400	...	Follow A069275 in OEIS
15th order	32768	49152	73728	81920	110592	114688	122880	165888	172032	180224	184320	204800	...	Follow A069276 in OEIS
16th order	65536	98304	147456	163840	221184	229376	245760	331776	344064	360448	368640	409600	...	Follow A069277 in OEIS
17th order	131072	196608	294912	327680	442368	458752	491520	663552	688128	720896	737280	819200	...	Follow A069278 in OEIS
18th order	262144	393216	589824	655360	884736	917504	983040	1327104	1376256	1441792	1474560	1638400	...	Follow A069279 in OEIS
19th order	524288	786432	1179648	1310720	1769472	1835008	1966080	2654208	2752512	2883584	2949120	3276800	...	Follow A069280 in OEIS
20th order	1048576	1572864	2359296	2621440	3538944	3670016	3932160	5308416	5505024	5767168	5898240	6553600	...	Follow A069281 in OEIS

properties

• Every prime is a near-prime of order 1, and every composite number is a near-prime of order 2 or higher. Third-order almost prime numbers, provided they consist of 3 different prime factors, are also called sphenic numbers .
• The union of the form a decomposition of the natural numbers. ${\displaystyle P_{n}}$
• Every -th order almost primes is the product of the orders of almost primes with , e.g. B .: The product of the 3-fast prime number 12 and the 4-fast prime number 40 results in the 7-fast prime number 480. There are such possible decompositions for, where the Stirling numbers denote the second kind. ${\displaystyle n}$ ${\displaystyle k_{1},\dotsc ,k_{m}}$ ${\displaystyle \sum _{i=1}^{m}k_{i}=n}$ ${\displaystyle k_{1},\dotsc ,k_{m}>0}$ ${\displaystyle S_{n,m}}$ ${\displaystyle S_{n,m}}$
• Since there is no possible prime factorization for zero , it is not an almost- th order prime . ${\displaystyle n}$
• The One is the empty product assigned as prime factorization. Correspondingly, according to the definition, it can be referred to as an almost prime number of the 0th order.
• Let the number of positive integers be less than or equal to exactly prime divisors (which do not necessarily have to be different). Then: ${\displaystyle \pi _{k}(n)}$ ${\displaystyle n}$ ${\displaystyle k}$
  ${\displaystyle \pi _{k}(n)\sim {\frac {n}{\log n}}\cdot {\frac {(\log \log n)^{k-1}}{(k-1)!}}}$
• Any sufficiently large even number can be represented as the sum of a prime number and a second-order near-prime number.
  This statement is similar to Goldbach's Hypothesis , was proven by Chen Jingrun in 1978 and is called Chen's theorem .
• There are infinitely many prime numbers, so is a 2-near prime number. This statement is similar to the conjecture about prime twins and was also proven by Chen. ${\displaystyle p+2}$
  

Applications

Second order almost primes, i.e. products of two prime numbers, are used in cryptography .

Web links

• Eric W. Weisstein : Almost prime . In: MathWorld (English). Near prime numbers
• Eric W. Weisstein : Semiprime . In: MathWorld (English). 2nd order almost primes

literature

• Władysław Narkiewicz : The Development of Prime Number Theory. From Euclid to Hardy and Littlewood. Springer, Berlin a. a. 2000, ISBN 3-540-66289-8 .
• Hans Riesel : Prime Numbers and Computer Methods for Factorization. Birkhäuser, Boston / Basel / Stuttgart 1985, ISBN 3-7643-3291-3 .
• David M. Bressoud: Factorization and Primality Testing. Springer, New York a. a. 1989, ISBN 0-387-97040-1 .
• Paulo Ribenboim : The little book of bigger primes. 2nd edition. Springer, New York a. a. 2004, ISBN 0-387-20169-6 .
• The world of prime numbers. Secrets and Records. Updated by Wilfrid Keller. Springer, Berlin / Heidelberg / New York 2006, ISBN 978-3-540-34283-0 .

Individual evidence

1. ↑ Wolfgang Blum: Goldbach and the twins. In: Spektrum der Wissenschaft , December 2008, p. 97 (reproduced: Prime numbers: Who will reveal the secret of indivisibility? Spiegel Online , December 25, 2008; accessed August 24, 2018).
2. ↑ Paulo Ribenboim: The world of prime numbers. Secrets and Records. Springer, Berlin / Heidelberg / New York 2006, ISBN 978-3-540-34283-0 , p. 219.
3. ↑ Edmund Landau : Handbook of the theory of the distribution of prime numbers. BG Teubner, 1909, p. 211 , accessed on June 30, 2018 . 
4. ↑ ^{a b} Konstantin Fackeldey: The Goldbach conjecture and its previous attempts at a solution. (PDF) Freie Universität Berlin , 2002, pp. 25–27 , accessed on June 30, 2018 .