Prime Number

A prime number (or prime integer, often simply called a "prime" for short) is a positive integer  that has no positive integer divisors other than 1 and  itself. More concisely, a prime number  is a positive integer having exactly one positive divisor other than 1, meaning it is a number that cannot be factored. For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while the number 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization ), making 24 not a prime number. Positive integers other than 1 which are not prime are called composite numbers.

While the term "prime number" commonly refers to prime positive integers, other types of primes are also defined, such as the Gaussian primes.

The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909, 1914; Hardy and Wright 1979, p. 11; Gardner 1984, pp. 86-87; Sloane and Plouffe 1995, p. 33; Hardy 1999, p. 46), it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. A good reason not to call 1 a prime number is that if 1 were prime, then the statement of the fundamental theorem of arithmetic would have to be modified since "in exactly one way" would be false because any . In other words, unique factorization into a product of primes would fail if the primes included 1. A slightly less illuminating but mathematically correct reason is noted by Tietze (1965, p. 2), who states "Why is the number 1 made an exception? This is a problem that schoolboys often argue about, but since it is a question of definition, it is not arguable." As more simply noted by Derbyshire (2004, p. 33), "2 pays its way [as a prime] on balance; 1 doesn't."

With 1 excluded, the smallest prime is therefore 2. However, since 2 is the only even prime (which, ironically, in some sense makes it the "oddest" prime), it is also somewhat special, and the set of all primes excluding 2 is therefore called the "odd primes." Note also that while 2 is considered a prime today, at one time it was not (Tietze 1965, p. 18; Tropfke 1921, p. 96).

The th prime number is commonly denoted , so , , and so on, and may be computed in the Wolfram Language as Prime[n].

The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ... (OEIS A000040; Hardy and Wright 1979, p. 3). A mnemonic for remembering the first seven primes is, "In the early morning, astronomers spiritualized nonmathematicians" (G. L. Honaker, Jr., pers. comm., Aug. 4, 2005). In the novel The Curious Incident of the Dog in the Night-Time (Haddon 2003), the protagonist Christopher amusingly numbers the chapters using the prime numbers instead of the (much) more traditional positive integers. In the Season 1 episode "Prime Suspect" (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes realized that character Ethan's daughter has been kidnapped because he is close to solving the Riemann hypothesis, which allegedly would allow the perpetrators to break essentially all internet security by factoring large numbers.

The numbers of decimal digits in  for , 1, ... is given by 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, ... (OEIS A099260).

The set of primes is sometimes denoted , represented in the Wolfram Language as Primes.

The first few primes are illustrated above as a sequence of binary bits.

Euler commented "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate" (Havil 2003, p. 163). In a 1975 lecture, D. Zagier commented "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision" (Havil 2003, p. 171).

The th prime for , 1, ... is given by 2, 29, 541, 7919, 104729, 1299709, 15485863, 179424673, 2038074743, ... (OEIS A006988; Graham et al. 1990, p. 111).

Large primes (Caldwell) include the large Mersenne primes, Ferrier's prime, and the -digit counterexample  showing that 5359 is not a Sierpiński number of the second kind (Helm and Norris). The largest known prime as of December 2018 is the Mersenne prime , which has a whopping  decimal digits.

Prime numbers can be generated by sieving processes (such as the sieve of Eratosthenes), and lucky numbers, which are also generated by sieving, appear to share some interesting asymptotic properties with the primes. Prime numbers satisfy many strange and wonderful properties. Although there exist explicit prime formulas (i.e., formulas which either generate primes for all values or else the th prime as a function of ), they are contrived to such an extent that they are of little practical value.

The Dirichlet generating function of the characteristic function of the prime numbers  is given by

			(1)
			(2)
			(3)

where  is the prime zeta function and  is an Iverson bracket.

The function that gives the number of primes less than or equal to a number  is denoted  and is called the prime counting function. The theorem giving an asymptotic form for  is called the prime number theorem. Similarly, the numbers of primes of the form  less than or equal to a number  is denoted  and is called the modular prime counting function.

 and  are inverse functions, so

(4)

for all positive integers and

(5)

iff  is a prime number.

Many prime factorization algorithms have been devised for determining the prime factors of a given integer, a process known as factorization or prime factorization. They vary quite a bit in sophistication and complexity. It is very difficult to build a general-purpose algorithm for this computationally "hard" problem, so any additional information which is known about the number in question or its factors can often be used to save a large amount of time. It should be emphasized that although no efficient algorithms are known for factoring arbitrary integers, it has not been proved that no such algorithm exists. It is therefore conceivable that a suitably clever person could devise a general method of factoring which would render the vast majority of encryption schemes in current widespread use, including those used by banks and governments, easily breakable.

Because of their importance in encryption algorithms such as RSA encryption, prime numbers can be important commercial commodities. In fact, R. Schlafly (1994) has obtained U.S. Patent  on the following two primes (expressed in hexadecimal notation):

(6)

and

(7)

The fundamental theorem of arithmetic states that any positive integer can be represented in exactly one way as a product of primes. Euclid's second theorem demonstrated that there are an infinite number of primes. However, it is not known if there are an infinite number of primes of the form  (Hardy and Wright 1979, p. 19; Ribenboim 1996, pp. 206-208), whether there are an infinite number of twin primes (the twin prime conjecture), or if a prime can always be found between  and  (Hardy and Wright 1979, p. 415; Ribenboim 1996, pp. 397-398). The latter two of these are two of Landau's problems.

The simplest method of finding factors is so-called "direct search factorization" (a.k.a. trial division). In this method, all possible factors are systematically tested using trial division to see if they actually divide the given number. It is practical only for very small numbers. More general (and complicated) methods include the elliptic curve factorization method and number field sieve factorization method.

It has been proven that the set of prime numbers is a Diophantine set (Ribenboim 1991, pp. 106-107).

With the exception of 2 and 3, all primes are of the form , i.e.,  (Bungus 1599, p. 399, quoted in Peano 1908, p. 59; Wells 1986, p. 68). For  an integer ,  is prime iff the congruence equation

(8)

holds for , 1, ...,  (Deutsch 1996), where  is a binomial coefficient. In addition, an integer  is prime iff

(9)

The first few composite  for which  are , 560, 588, 1400, 23760, ... (OEIS A011774; Guy 1997), with a total of 18 such numbers less than .

Chen (1979) showed that for  sufficiently large, there always exists a number with at least two prime factors between  and  for  (Le Lionnais 1983, p. 26; Guy 2004, p. 34). In practice, this relation seems to hold for all .

Primes consisting of consecutive digits (counting 0 as coming after 9) include 2, 3, 5, 7, 23, 67, 89, 4567, 78901, ... (OEIS A006510). Primes consisting of digits that are themselves primes include 23, 37, 53, 73, 223, 227, 233, 257, 277, 337, 353, 373, 523, 557, ... (OEIS A019546), which is one of the Smarandache sequences.

Because a prime number  has only the trivial factors 1 and , in his The Road Ahead, Bill Gates accidentally referred to a trivial operation when he stated "Because both the system's privacy and the security of digital money depend on encryption, a breakthrough in mathematics or computer science that defeats the cryptographic system could be a disaster. The obvious mathematical breakthrough would be the development of an easy way to factor large prime numbers [emphasis added]" (Gates 1995, p. 265).

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