 A '''prime number''' (or a '''prime''') is a natural number that has exactly two ''distinct'' natural number divisors: 1 and itself. The smallest twenty-five prime numbers (all the prime numbers under 100) are: : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 . An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC, although the density of prime numbers within natural numbers is 0. The number 1 is by definition not prime. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any positive integer  can be factored into primes, written as a product of primes or powers of different primes (including the empty product of factors for 1). Moreover, this factorization is unique except for a possible reordering of the factors.  The property of being prime is called '''primality'''. Verifying the primality of a given number  can be done by trial division. The simplest trial division method tests whether  is a multiple of an integer ''m'' between 2 and&amp;nbsp;. If  is a multiple of any of these integers then it is a composite number, and so not prime; if it is not a multiple of any of these integers then it is prime. As this method requires up to  trial divisions, it is only suitable for relatively small values of&amp;nbsp;. More sophisticated algorithms, which are much more efficient than trial division, have been devised to test the primality of large numbers.  There is no known useful formula that yields all of the prime numbers and no composites. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large can be modeled. The first result in that direction is the prime number theorem which says that the probability that a given, randomly chosen number  is prime is inversely proportional to its number of digits, or the logarithm of&amp;nbsp;.  This statement has been proven since the end of the 19th century. The unproven Riemann hypothesis dating from 1859 implies a refined statement concerning the distribution of primes.  Despite being intensely studied, there remain some open questions around prime numbers which can be stated simply. For example, Goldbach's conjecture, which asserts that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, which says that there are infinitely many twin primes (pairs of primes whose difference is 2), have been unresolved for more than a century, notwithstanding the simplicity of their statements.  Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, notably the notion of prime ideals.  Primes are applied in several routines in information technology, such as public-key cryptography, which makes use of the difficulty of factoring large numbers into their prime factors. Searching for big primes, often using distributed computing, has stimulated studying special types of primes, chiefly Mersenne primes whose primality is comparably quick to decide. , the largest known prime number has about 13 million decimal digits.GIMPS Home;   ==Definition and examples== A natural number :1, 2, 3, 4, 5, 6, ... is called a '''prime''' or a '''prime number''' if it is greater than 1 and has exactly two divisors, 1 and the number itself., p. 10, section 2 Natural numbers greater than 1 that are not prime are called ''composite''.   Among the numbers 1 to 6, the numbers 2, 3, and 5 are the prime numbers, while 1, 4, and 6 are not prime. 1 is excluded as a prime number, for reasons explained below. 2 is a prime number, since the only natural numbers dividing it are 1 and 2. Next, 3 is prime, too: 1 and 3 do divide 3 without remainder, but 3 divided by 2 gives remainder 1. Thus, 3 is prime. However, 4 is composite, since 2 is another number (in addition to 1 and 4) dividing 4 without remainder: :4 = 2 &amp;middot; 2. 5 is again prime: none of the numbers 2, 3, or 4 divide 5. Next, 6 is divisible by 2 or 3, since :6 = 2 &amp;middot; 3. Hence, 6 is not prime.  The image at the right illustrates that 12 is not prime: 12 = 3 &amp;middot; 4. More generally, no even number bigger than 2 is prime: any such number ''n'' has at least three distinct divisors, namely 1, 2, and ''n''. This implies that ''n'' is not prime. Accordingly, the term ''odd prime'' refers to any prime number greater than 2. In a similar vein, all prime numbers bigger than 5, written in the usual decimal system, end in 1, 3, 7 or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.  For any natural number , 1 and  divide  without remainder. Therefore, the condition on being a prime can also be restated as: a number is prime if it is bigger than one and if none of : divides  (without remainder). Yet another way to say the same is: a number  is prime if it cannot be written as a product of two integers  and , both of which are larger than&amp;nbsp;1: :.  The set of all primes is often denoted .  ==The fundamental theorem of arithmetic==  The crucial importance of prime numbers to number theory and mathematics in general stems from the ''fundamental theorem of arithmetic'', which states that every positive integer larger than 1 can be written as a product of one or more primes in a way which is unique except possibly for the order of the prime factors. Primes can thus be considered the “basic building blocks” of the natural numbers. For example:  :{| |- |23244 ||= 2 · 2 · 3 · 13 · 149 |- |||= 22 · 3 · 13 · 149. (22 denotes the square or second power of 2.) |}  As in this example, the same prime factor may occur multiple times. A decomposition:  :  of a number  into (finitely many) prime factors , , ... to  is called ''prime factorization'' of . The fundamental theorem of arithmetic can be rephrased so as to say that any factorization into primes will be identical except for the order of the factors. So, albeit there are many prime factorization algorithms to do this in practice for larger numbers, they all have to yield the same result.  If  is a prime number and  divides a product  of integers, then  divides  or  divides . This proposition is known as Euclid's lemma.  It is used in some proofs of the uniqueness of prime factorizations.  ===Primality of one===  Until the 19th century, most mathematicians considered the number 1 a prime. Then, the definition was that a prime is divisible only by 1 and itself. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime.  Although a large body of mathematical work is also valid when calling 1 a prime, the above fundamental theorem of arithmetic does not hold as stated. For example, the number 15 can be factored as  or . If 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.&quot;[ &quot;Arguments for and against the primality of 1]&quot;.  ==History==  is a simple algorithm for finding all prime numbers up to a specified integer. It was created in the 3rd century BC by Eratosthenes, an ancient Greek mathematician.  (Click to see animation.)]] There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.  After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). A special case of Fermat's theorem may have been known much earlier by the Chinese.  Fermat conjectured that all numbers of the form 22''n''&amp;nbsp;+&amp;nbsp;1 are prime (they are called Fermat numbers) and he verified this up to ''n''&amp;nbsp;=&amp;nbsp;4 (or 216&amp;nbsp;+&amp;nbsp;1). However, the very next Fermat number 232&amp;nbsp;+&amp;nbsp;1 is composite (one of its prime factors is&amp;nbsp;641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime.  The French monk Marin Mersenne looked at primes of the form 2''p''&amp;nbsp;&amp;minus;&amp;nbsp;1, with ''p'' a prime.  They are called Mersenne primes in his honor.  Euler's work in number theory included many results about primes.  He showed the infinite series  is divergent. In 1747 he showed that the even perfect numbers are precisely the integers of the form 2''p''&amp;minus;1(2''p''&amp;nbsp;&amp;minus;&amp;nbsp;1), where the second factor is a Mersenne prime.  At the start of the 19th century, Legendre and Gauss independently conjectured that as ''x'' tends to infinity, the number of primes up to ''x'' is asymptotic to ''x''/ln(''x''), where ln(''x'') is the natural logarithm of ''x''.   Ideas of Riemann in his 1859 paper on the zeta-function sketched a program which would lead to a proof of the prime number theorem. This outline was completed by Hadamard and de la Vallée Poussin, who independently proved the prime number theorem in 1896.  Proving a number is prime is not done (for large numbers) by trial division. Many mathematicians have worked on primality tests for large numbers, often restricted to specific number forms. This includes Pépin's test for Fermat numbers (1877), Proth's theorem (around 1878), the Lucas–Lehmer primality test (originated 1856),[ The Largest Known Prime by Year: A Brief History] [ Prime Curios!: 17014…05727 (39-digits)] and the generalized Lucas primality test. More recent algorithms like APRT-CL, ECPP, and AKS work on arbitrary numbers but remain much slower.  For a long time, prime numbers were thought to have extremely limited application outside of pure mathematics; this changed in the 1970s when the concepts of public-key cryptography were invented, in which prime numbers formed the basis of the first algorithms such as the RSA cryptosystem algorithm.  Since 1951 all the largest known primes have been found by computers. The search for ever larger primes has generated interest outside mathematical circles. The Great Internet Mersenne Prime Search and other distributed computing projects to find large primes have become popular in the last ten to fifteen years, while mathematicians continue to struggle with the theory of primes.  ==The number of prime numbers==  There are infinitely many prime numbers. Another way of saying this is that the sequence :2, 3, 5, 7, 11, 13, ... of prime numbers never ends. This statement is referred to as ''Euclid's theorem'' in honor of the ancient Greek mathematician Euclid, since the first known proof for this statement is attributed to him. Many more proofs of the infinitude of primes are known, including an analytical proof by Euler, Goldbach's proof based on  Fermat numbers,[ Letter] in Latin from Goldbach to Euler, July 1730. Fürstenberg's proof using general topology, and Kummer's elegant proof.  ===Euclid's proof===  Euclid's proof (Book IX, Proposition 20James Williamson (translator and commentator), ''The Elements of Euclid, With Dissertations'', Clarendon Press, Oxford, 1782, page 63.) begins with the definition of prime and then considers any finite set of primes, which we denote ''p''1, ''p''2, up to ''p''''n''.  The key idea is to consider the product of all these numbers plus one (this number is called a Euclid number):  :''P'' = ''p''1 &amp;middot; ''p''2 &amp;middot; ... &amp;middot; ''p''''n'' + 1.  Like any natural number, ''P'' can be written as a product of prime numbers; this is assured by the fundamental theorem of arithmetic:  :''P'' = ''q''1 &amp;middot; ''q''2 &amp;middot; ... &amp;middot; ''q''''m''  (it is possible that ''P'' itself is prime, in which case ''m''&amp;nbsp;=&amp;nbsp;1).  None of the primes ''p''1, ''p''2, etc., to ''p''''n'' can divide ''P'', because dividing ''P'' by any of these leaves a remainder of 1.  Therefore the primes ''q''1, ''q''2, ..., ''q''''m'' are additional primes beyond the ones we started with.  Thus any finite set of primes can be extended to a larger finite set of primes.  The Euclid number ''P'' need not be prime, for example :2 &amp;middot; 3 &amp;middot; 5 &amp;middot; 7 &amp;middot; 11 &amp;middot; 13 + 1 = 30,031 = 59 &amp;middot; 509 (both primes).  It is often erroneously reported that Euclid proved this result by contradiction, beginning with the assumption that the set initially considered contains all prime numbers, or that it contains precisely the ''n'' smallest primes, rather than any arbitrary finite set of primes.Michael Hardy and Catherine Woodgold, &quot;Prime Simplicity&quot;, ''Mathematical Intelligencer'', volume 31, number 4, fall 2009, pages 44–52.  ===Euler's analytical proof=== Euler's proof uses the sum of the reciprocals of primes,  :S(p) = \frac 1 2 + \frac 1 3 + \frac 1 5 + \frac 1 7 + \cdots + \frac 1 p.  This sum becomes bigger than any arbitrary real number provided that ''p'' is big enough., Section 1.6, Theorem 1.13 This shows that there are infinitely many primes, since otherwise this sum would grow only until the biggest prime ''p'' is reached. The sum ''S''(''p'') grows as fast as ln(ln(''p'')), up to a bounded error term (Mertens' second theorem). For comparison, the sums  :\frac 1 {1^2} + \frac 1 {2^2} + \frac 1 {3^2} + \cdots + \frac 1 {n^2} = \sum_{i=1}^n \frac 1 {i^2}  do not grow to infinity, when ''n'' grows. In this sense, prime numbers occur more often than squares of natural numbers. Brun's theorem states that the sum of the reciprocals of twin primes,  : \left( {\frac{1}{3} + \frac{1}{5}} \right) + \left( {\frac{1}{5} + \frac{1}{7}} \right) + \left( {\frac{1} + \frac{1}} \right) +  \cdots = \sum\limits_{ \begin{smallmatrix} p \text{ prime, } \\ p + 2 \text { prime} \end{smallmatrix}} {\left( {\frac{1}{p} + \frac{1}} \right)},  is finite.  ==Testing primality and integer factorization==  Various methods called ''primality tests'' determine whether a given number ''n'' is prime. Most such methods only tell whether ''n'' is prime or not but do not yield the prime factors of ''n''. A routine accomplishing the latter task, too, is called a ''factorization algorithm''.  ===Trial division=== The most basic method of checking the primality of a given integer ''n'' is called ''trial division''. This routine consists in dividing ''n'' by each integer ''m'' which is greater than 1 and less than or equal to the square root of ''n''. If the result of any of these divisions is an integer, then ''n'' is not a prime, otherwise, it is a prime. Indeed, if ''n'' = ''ab'' with both ''a'' and ''b'' strictly smaller than ''n'', ''a'' or ''b'' (or both) are necessarily smaller than √. For example, for  35}}, the procedure divides 37 by  2, 3, 4, 5, and 6.}} Since none of these numbers divides 37, trial division shows that 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to √ is known&amp;mdash;then trial divisions only need to be checked for those ''m'' that are prime. For example, to check the primality of 37, only three divisions are necessary (''m'' = 2, 3, and 5), given that 4 and 6 are composite.  While a simple method, trial division quickly becomes impractical for testing large integers because the number of possible factors grows too rapidly as ''n'' increases. According to the prime number theorem explained below, the number of prime numbers less than √ is approximately given by  / ln(√)}}, so the algorithm may need up to this number of trial divisions to check the primality of ''n''. For  1020}}, this number is 450 million&amp;mdash;too large for many practical applications.  ===Sieves=== An algorithm yielding all primes up to a given limit, such as required in the trial division method, is called a sieve. The oldest example, the sieve of Eratosthenes (see above) is useful for relatively small primes. The modern sieve of Atkin is more complicated, but faster when properly optimized. Before the advent of computers, lists of primes up to bounds like 107 were also used..  ===Primality testing vs. primality proving=== Modern primality tests can be divided into two main classes, probabilistic (or &quot;Monte Carlo&quot;) and deterministic algorithms. The former merely &quot;test&quot; whether a given number ''n'' is prime in the sense that they declare ''n'' to be (definitely) composite or &quot;probably prime&quot;, which latter means that ''n'' may or may not be a prime number. Composite numbers which do pass a given primality test are referred to as pseudoprimes. For example, Fermat's primality test relies on Fermat's little theorem. This theorem says for any prime number ''p'',  is divisible by ''p''. Thus, if  is not divisible by ''n'', ''n'' cannot be prime. However, conversely, ''n'' may be composite even if this divisibility holds. In fact, there are infinitely many composite numbers ''n'' which pass the Fermat primality test no matter what choice of ''a'' is made (Carmichael numbers), for example  561}}.  Deterministic algorithms do not erroneously report composite numbers as prime. In practice, the fastest such method is known as elliptic curve primality proving. Analyzing its run time is based on heuristic arguments, as opposed to the rigorously proven complexity of the more recent AKS primality test. Deterministic methods are typically slower than probabilistic ones, so the latter ones are typically applied first before a more time-consuming deterministic routine is employed.  The following table lists a number of prime tests. The running time is given in terms of ''n'', the number to be tested and, for probabilistic algorithms, the number ''k'' of tests performed. Moreover, &amp;epsilon; is an arbitrarily small positive number, and log is the logarithm to an unspecified base. The big O notation means that, for example, elliptic curve primality proving requires a time that is bounded by a factor (not depending on ''n'', but on &amp;epsilon;) times log5+ε(''n'').  {| class=&quot;wikitable sortable&quot; |- ! Test ! Developed in ! Type ! Running time ! Notes |- | AKS primality test | 2002 | deterministic | O(log6+ε(''n'')) | |- | Elliptic curve primality proving | 1977 | deterministic | O(log5+ε(''n'')) ''heuristically'' | |- | Miller–Rabin primality test | 1980 | probabilistic | O(''k'' · log2+ε (''n'')) | error probability 4&amp;minus;''k'' |- | Solovay–Strassen primality test | 1977 | probabilistic | O(''k'' · log3 ''n'') | error probability 2&amp;minus;''k'' |- | Fermat primality test | | probabilistic | O(''k'' · log2+ε (''n'')) | fails for Carmichael numbers |}  ===Special-purpose algorithms===  thumb In addition to the aforementioned tests applying to any natural number ''n'', a number of much more efficient primality tests is available for special numbers. For example, to run Lucas' primality test requires the knowledge of the prime factors of , while the Lucas–Lehmer primality test needs the prime factors of  as input. For example, these tests can be applied to check whether :''n''! ± 1 = 1 · 2 · 3 · ... · ''n'' ± 1 are prime. Prime numbers of this form are known as factorial primes. Other primes where either ''p'' + 1 or ''p'' &amp;minus; 1 is of a particular shape include the Sophie Germain primes (primes of the form 2''p'' + 1 with ''p'' prime), primorial primes, Fermat primes and Mersenne primes, that is, prime numbers that are of the form , where ''p'' is an arbitrary prime. It is not known whether there are infinitely many primes of any of these types.  Fermat primes are of the form , with ''k'' an arbitrary natural number. They are named after Pierre de Fermat who conjectured that ''all'' numbers  22''k'' + 1}} are prime. This was based on the evidence of the first five numbers in this series&amp;mdash;3, 5, 17, 257, and 65,537&amp;mdash;being prime. However, ''F''6 is composite and so are all other Fermat numbers that have been verified as of 2011. A regular ''n''-gon is constructible using straightedge and compass if and only if :''n'' = 2''i'' &amp;middot; ''m'' where ''m'' is a product of any number of distinct Fermat primes and ''i'' is any natural number, including zero.  ===The largest known prime===  Since the dawn of electronic computers the largest known prime has almost always been a Mersenne prime because there is a fast algorithm, the Lucas–Lehmer primality test, for these numbers. Some of these primes have been found using distributed computing, such as the Great Internet Mersenne Prime Search. On October 22, 2009, this project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively. Other projects searching primes are PrimeGrid and Wieferich@Home, which tries to find the third Wieferich prime.  The following table gives the largest known primes of the mentioned types. Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256''k''''n'' + 1, 256''k''(''n'' + 1) &amp;minus; 1].  {| class=&quot;wikitable&quot; |- ! Prime ! Number of decimal digits ! Type ! Date ! Found by |- | 243,112,609 − 1 | style=&quot;text-align:right;&quot;| 12,978,189 | Mersenne prime | August 23, 2008 | Great Internet Mersenne Prime Search |- | 19,249 × 213,018,586 + 1 | style=&quot;text-align:right;&quot;| 3,918,990 | not a Mersenne prime (Proth number) | March 26, 2007 | Seventeen or Bust |- | 94550! − 1 | style=&quot;text-align:right;&quot;| 429,390 | factorial prime | October 2010 | Domanov, PrimeGrid |- | 843301# - 1 | style=&quot;text-align:right;&quot;| 365,851 | primorial prime | December 2010 | PrimeGrid[ Primegrid.com]; official anouncement, 24 December 2010 |- | 65516468355 × 2333333 ± 1 | style=&quot;text-align:right;&quot;| 100,355 | twin primes | 2009 | Twin prime search |}  ===Integer factorization===  Given a composite integer ''n'', the task of providing one (or all) prime factors is referred to as ''factorization'' of ''n''. Elliptic curve factorization is an algorithm relying on arithmetic on an elliptic curve.  ==Generating prime numbers==   There are many formulas for primes, but they are usually computationally inefficient. For example, Mills' theorem asserts that there is a number ''A'' such that the floor of (i.e., smallest integer not greater than) ''A''3''n'' is a prime for any natural number ''n''. However, computing such a number ''A'' requires the knowledge of infintely many primes to begin with.  One formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.  There are other similar formulas which also produce primes.  There is no non-constant polynomial, even in several variables, that takes only prime values.  But there is a set of Diophantine equations in 9 variables and one parameter with the following property: the parameter is prime if and only if the resulting system of equations has a solution over the natural numbers. This can be used to obtain a single formula with the property that all its ''positive'' values are prime.  ==Distribution== The [[Ulam spiral. Black pixels show prime numbers.|200px|thumb]] The problem of modeling the distribution of prime numbers is a popular subject of investigation for number theorists. In a 1975 lecture, noted number theorist Don Zagier commented that primes both &quot;grow like weeds among the natural numbers, seeming to obey no other law than that of chance&quot; but also &quot;exhibit stunning regularity&quot; and &quot;that there are laws governing their behavior, and that they obey these laws with almost military precision.  Euler noted that the function :''n''2 + ''n'' + 41 gives prime numbersSee [ list of values], calculated by Wolfram Alpha for ''n'' Hua (2009), &quot; a remarkable fact leading into deep algebraic number theory, more specifically Heegner numbers. The Ulam spiral depicts all natural numbers in a spiral-like way. Surprisingly, prime numbers cluster on certain diagonals and not others.  Bertrand's postulate, proven first by Chebyshev, states that there always exists at least one prime number ''p'' with ''n''&amp;nbsp;&amp;nbsp;3  ===Number of prime numbers below a given number===  A chart depicting &amp;pi;(''n'') (blue), ''n'' / ln (''n'') (green) and Li (''n''), the [[offset logarithmic integral (red)|right|thumb|200px]] The ''prime counting function'' π(''n'') is defined as the number of primes up to ''n''. For example π(11) = 5, since there are five primes less than or equal to 11. There are known algorithms to compute exact values of π(''n'') faster than it would be possible to compute each prime up to ''n''. The prime number theorem states that π(''n'') is approximately given by :\pi(n) \approx \frac n {\ln(n)}. In other words, as ''n'' gets very large, the likelihood that a number less than ''n'' is prime is inversely proportional to the number of digits in ''n''. A more accurate estimate is given by the offset logarithmic integral.  ===Gaps between primes===  A sequence of consecutive integers none of which is prime constitutes a ''prime gap''. There are arbitrarily long prime gaps: the numbers  (for the notation ''n''! read factorial) is a sequence of  consecutive composite integers. On the other hand, the gaps get arbitrarily small in proportion to the primes: the quotient :\frac{p_{i+1} - p_i}{p_i},  where ''p''''i'' denotes the ''i''th prime number (i.e., ''p''1 = 2, ''p''2 = 3, etc.), approaches zero as ''i'' approaches infinity.  ===Arithmetic progressions===  An arithmetic progression is the set of natural numbers that give the same remainder when divided by some fixed number&amp;nbsp;''q'' called modulus. For example, :3, 12, 21, 30, 39, ..., is an arithmetic progression modulo  9}}. Except for 3, none of these numbers is prime, since  3(1 + 3''n'')}} so that the remaining numbers in this progression are all composite. (More generally, all prime numbers above ''q'' are of form ''q''#·''n''&amp;nbsp;+&amp;nbsp;''m'', where 0&amp;nbsp;|title=The primes contain arbitrarily long arithmetic progressions|journal=Annals of Mathematics|volume=167|year=2008|pages=481–547}}. An odd prime ''p'' is expressible as the sum of two squares,  ''x''2 + ''y''2}}, exactly if ''p'' is congruent 1 modulo 4 (Fermat's theorem on sums of two squares).  ==Open questions== ===The zeta function and the Riemann hypothesis===  Plot of the zeta function &amp;zeta;(''s''). At ''s''=1, the function has a [[pole, i.e. tends to infinity.]]  The Riemann zeta function &amp;zeta;(''s'') is defined as an infinite sum :\zeta(s)=\sum_{n=1}^\infin \frac{1}{n^s}, where ''s'' is a complex number with real part bigger than 1. It is a consequence of the fundamental theorem of arithmetics that this sum agrees with the infinite product :\prod_{p \text{ prime}} \frac{1}{1-p^{-s}}. The zeta function is closely related to prime numbers. For example, the afore-mentioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then &amp;zeta;(1) would have a finite value. However, the harmonic series  diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler, :\zeta(2) = \prod_{p} \frac{1}{1-p^{-2}}= \frac{\pi^2}{6}. The reciprocal of &amp;zeta;(2), 6/&amp;pi;2, is the probability that two numbers selected at random are relatively prime.C. S. Ogilvy &amp; J. T. Anderson ''Excursions in Number Theory'', pp. 29–35, Dover Publications Inc., 1988 ISBN 0-486-25778-9  The ''Riemann hypothesis'' states that, except for  &amp;minus;2, &amp;minus;4, ...,}} all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible. From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about 1/ log ''x'' of numbers less than ''x'' are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of ''x'' (for intervals near ''x''). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.  ===Other conjectures===  In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture which asserts that every even integer ''n'' greater than 2 can be written as a sum of two primes. , this conjecture has been verified for all numbers up to  2 &amp;middot; 1017}}. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer greater can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.  Other conjectures deal with the question whether an infinity of prime numbers subject to certain constraints exists. It is conjectured that there are infinitely many Fibonacci primesCaldwell, Chris, [ ''The Top Twenty: Lucas Number''] at The Prime Pages. and infinitely many Mersenne primes, but not Fermat primes.E.g., see  It is not known whether or not there are an infinite number of prime Euclid numbers.  A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer ''n'', there are infinitely many pairs of consecutive primes that differ by&amp;nbsp;2''n''. It is conjectured there are infinitely many primes of the form&amp;nbsp;''n''2&amp;nbsp;+&amp;nbsp;1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between ''n''2 and (''n''&amp;nbsp;+&amp;nbsp;1)2 for every positive integer&amp;nbsp;''n''. It is implied by the stronger Cramér's conjecture.  ==Applications==  For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. &quot;No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years.&quot; However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators.  Some rotor machines were designed with a different number of pins on each rotor, with the number of pins on any one rotor either prime, or coprime to the number of pins on any other rotor. This helped generate the full cycle of possible rotor positions before repeating any position.  The International Standard Book Numbers work with a check digit, which exploits the fact that 11 is a prime.  ===Arithmetic modulo a prime and finite fields===  ''Modular arithmetic'' modifies usual arithmetic by only using the numbers :\{0, 1, 2, \dots, n-1 \}, \, where ''n'' is a fixed natural number called modulus. Calculating sums, differences and products is done as usual, but whenever a negative number or a number greater than ''n''&amp;minus;1 occurs, it gets replaced by the remainder after division by ''n''. For instance, for ''n''&amp;nbsp;=&amp;nbsp;7, the sum 3&amp;nbsp;+&amp;nbsp;5 is 1 instead of 8, since 8 divided by 7 has remainder&amp;nbsp;1. This is referred to by saying &quot;3 + 5 is congruent to 1 modulo 7&quot; and is denoted :3 + 5 \equiv 1 \ \ (\operatorname{mod}\ 7). Similarly, 6&amp;nbsp;+&amp;nbsp;1&amp;nbsp;≡&amp;nbsp;0&amp;nbsp;(mod&amp;nbsp;7), 2&amp;nbsp;&amp;minus;&amp;nbsp;5&amp;nbsp;≡&amp;nbsp;4&amp;nbsp;(mod&amp;nbsp;7), since &amp;minus;3&amp;nbsp;+&amp;nbsp;7&amp;nbsp;=&amp;nbsp;4, and 3&amp;nbsp;·&amp;nbsp;4&amp;nbsp;≡&amp;nbsp;5&amp;nbsp;(mod&amp;nbsp;7) as 12 has remainder 5. Standard properties of addition and multiplication  familiar from the integers remain valid in modular arithmetic. In the parlance of abstract algebra, the above set of integers, which is also denoted '''Z'''/''n'''''Z''', is therefore a commutative ring for any ''n''. Division, however, is not in general possible in this setting. For example, for ''n'' = 6, the equation :3 \cdot x \equiv 2 \ \ (\operatorname{mod}\ 6), a solution ''x'' of which would be an analogue of 2/3, cannot be solved, as one can see by calculating 3 · 0, ..., 3 · 5 modulo 6. The distinctive feature of prime numbers is the following: division ''is'' possible in modular arithmetic if and only if ''n'' is a prime. Equivalently, ''n'' is prime if and only if all integers ''m'' satisfying  are ''coprime'' to ''n'', i.e. their only common divisor is one. Indeed, for ''n'' = 7, the equation :3 \cdot x \equiv 2 \ \ (\operatorname{mod}\ 7), has a unique solution,  3}}. Because of this, for any prime ''p'', '''Z'''/''p'''''Z''' (also denoted '''F'''''p'') is called a field or, more specifically, a finite field since it contains finitely many, namely ''p'', elements.  A number of theorems can be derived from inspecting '''F'''''p'' in this abstract way. For example, Fermat's little theorem, stating :a^{p-1} \equiv 1 (\operatorname{mod}\ p) for any integer ''a'' not divisble by ''p'', may be proved using these notions. This implies :\sum_{a=1}^{p-1} a^{p-1} \equiv (p-1) \cdot 1 \equiv -1 \pmod p. Giuga's conjecture says that this equation is also a sufficient condition for ''p'' to be prime. Another consequence of Fermat's little theorem is the following: if ''p'' is a prime number other than 2 and 5, 1/''p'' is always a recurring decimal, whose period is  or a divisor of .  The fraction 1/''p'' expressed likewise in base ''q'' (rather than base&amp;nbsp;10) has similar effect, provided that ''p'' is not a prime factor of&amp;nbsp;''q''. Wilson's theorem says that an integer ''p''&amp;nbsp;&gt;&amp;nbsp;1 is prime if and only if the factorial (''p''&amp;nbsp;&amp;minus;&amp;nbsp;1)!&amp;nbsp;+&amp;nbsp;1 is divisible by ''p''. Moreover, an integer ''n'' &gt; 4 is composite if and only if (''n''&amp;nbsp;&amp;minus;&amp;nbsp;1)! is divisible by&amp;nbsp;''n''.  ===Other mathematical occurrences of primes=== Many mathematical domains make great use of prime numbers. An example from the theory of finite groups are the Sylow theorems: if ''G'' is a finite group and ''p''''n'' is the highest power of the prime ''p'' which divides the order of ''G'', then ''G'' has a subgroup of order ''p''''n''. Also, any group of prime order is cyclic (Lagrange's theorem).   ===Public-key cryptography===  Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example 512 bit primes are frequently used for RSA and 1024 bit primes are typical for Diffie–Hellman.). RSA relies on the fact that it is thought to be much easier (i.e., more efficient) to perform the multiplication of two (large) numbers ''x'' and ''y'' than to calculate ''x'' and ''y'' (assumed coprime) if only the product ''xy'' is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.  ===Prime numbers in nature=== Inevitably, some of the numbers that occur in nature are prime. There are, however, relatively few examples of numbers that appear in nature ''because'' they are prime.  One example of the use of prime numbers in nature is as an evolutionary strategy used by cicadas of the genus ''Magicicada''.Goles, E., Schulz, O. and M. Markus (2001). &quot;Prime number selection of cycles in a predator-prey model&quot;, Complexity 6(4): 33-38 These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on ''Magicicadas''. If ''Magicicadas'' appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.  There is speculation that the zeros of the zeta function are connected to the energy levels of complex quantum systems.  ==Generalizations== The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, &quot;prime&quot; indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field ''F'' containing both 0 and 1. It is either '''Q''' or the finite field with ''p'' elements, whence the name. | year=2002 | volume=211}}, Section II.1, p. 90 Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot which is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots.Schubert, H. &quot;Die eindeutige Zerlegbarkeit eines Knotens in Primknoten&quot;. ''S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl.'' 1949 (1949), 57–104. Prime models and prime 3-manifolds are other examples of this type.  ===Prime elements in rings===  Prime numbers give rise to two more general concepts that apply to elements of any commutative ring ''R'', an algebraic structure where addition, subtraction and multiplication are defined: ''prime elements'' and ''irreducible elements''. An element ''p'' of ''R'' is called prime element if it is neither zero nor a unit (i.e., does not have a multiplicative inverse) and satisfies the following requirement: given ''x'' and ''y'' in ''R'' such that ''p'' divides the product ''xy'', then ''p'' divides ''x'' or ''y''. An element is irreducible if it cannot be written as a product of two ring elements that are not units. In the ring '''Z''' of integers, the set of prime elements equals the set of irreducible elements, which is :\{ \dots, -11, -7, -5, -3, -2, 2, 3, 5, 7, 11, \dots \}\, . In any ring ''R'', any prime element is irreducible. The converse does not hold in general, but does hold for unique factorization domains.  The fundamental theorem of arithmetic continues to hold in unique factorization domains. An example of such a domain is the Gaussian integers '''Z'''[''i''], that is, the set of complex numbers of the form ''a''&amp;nbsp;+&amp;nbsp;''bi'' where ''i'' denotes the imaginary unit and ''a'' and ''b'' are arbitrary integers. Its prime elements are known as Gaussian primes. Not every prime (in '''Z''') is a Gaussian prime: in the bigger ring '''Z'''[''i''], 2 factors into the product of the two Gaussian primes (1&amp;nbsp;+&amp;nbsp;''i'') and (1&amp;nbsp;−&amp;nbsp;''i''). Rational primes (i.e. prime elements in '''Z''') of the form 4''k''&amp;nbsp;+&amp;nbsp;3 are Gaussian primes, whereas rational primes of the form 4''k''&amp;nbsp;+&amp;nbsp;1 are not.  ===Prime ideals===  In ring theory, the notion of number is generally replaced with that of ideal. ''Prime ideals'', which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), … The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem which expresses any ideal in a Noetherian commutative ring as the intersection of primary ideals, which are the appropriate generalizations of prime powers.  Prime ideals are the points of algebro-geometric objects, via the notion of the spectrum of a ring.Shafarevich, Basic Algebraic Geometry volume 2 (Schemes and Complex Manifolds), p. 5, section V.1 Arithmetic geometry also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or ramification of prime ideals when lifted to an extension field, a basic problem of algebraic number theory, bears some resemblance with ramification in geometry. Such ramification questions occur even in number-theoretic questions solely concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement which concerns the solvability of quadratic equations :x^2 \equiv p \ \ (\text{mod } q),\, where ''x'' is an integer and ''p'' and ''q'' are (usual) prime numbers.Neukirch, Algebraic Number theory, p. 50, Section I.8 Early attempts to prove Fermat's Last Theorem climaxed when Kummer introduced regular primes, primes satisfying a certain requirement concerning the failure of unique factorization in the ring consisting of expressions :a_0 + a_1 \zeta + ... + a_{p-1} \zeta^{p-1}\, ,  where ''a''0, ..., ''a''''p''&amp;minus;1 are integers and &amp;zeta; is a complex number such that  1}}.Neukirch, Algebraic Number theory, p. 38, Section I.7  ===Valuations=== Valuation theory studies certain functions from a field ''K'' to the real numbers '''R''' called valuations.Endler, Valuation Theory, p. 1  Every such valuation yields a topology on ''K'', and two valuations are called equivalent if they yield the same topology. A ''prime of K'' (sometimes called a ''place of K'') is an equivalence class of valuations. For example, the ''p''-adic valuation of a rational number ''q'' is defined to be the integer ''v''''p''(''q''), such that :q = p^{v_p(q)} \frac {r}{s}, where both ''r'' and ''s'' are not divisible by ''p''. For example,  2.}} The ''p''-adic norm is defined as :\left| \frac a b \right|_p := e^{-v_p(a/b)}. \, Thus, in contrast to the usual absolute value (also referred to as the infinite prime), the norm gets smaller when a number is multiplied by ''p''. While completing '''Q''' (roughly, filling the gaps) with respect to the absolute value yields the field of real numbers, completing with respect to  |&amp;minus;|''p'' yields '''Q'''''p'', the field of ''p''-adic numbers.Gouvea: p-adic numbers: an introduction, Chapter 3, p. 43 These are essentially all possible ways to complete '''Q''', by Ostrowski's theorem. Certain arithmetic questions related to '''Q''' or more general global fields may be transferred back and forth to the completed (or local) fields. This local-global principle again underlines the importance of primes to number theory.  ==In the arts and literature== Prime numbers have influenced many artists and writers.  The French composer Olivier Messiaen used prime numbers to create ametrical music through &quot;natural phenomena&quot;. In works such as ''La Nativité du Seigneur'' (1935) and ''Quatre études de rythme'' (1949–50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47  and 53 appear in one of the études. According to Messiaen this way of composing was &quot;inspired by the movements of nature, movements of free and unequal durations&quot;.  In his science fiction novel ''Contact'', later made into a film of the same name, the NASA scientist Carl Sagan suggested that prime numbers could be used as a means of communicating with aliens, an idea that he had first developed informally with American astronomer Frank Drake in 1975.Carl Pomerance, [ Prime Numbers and the Search for Extraterrestrial Intelligence], Retrieved on December 22, 2007  Many films reflect a popular fascination with the mysteries of prime numbers and cryptography: films such as ''Cube'', ''Sneakers'', ''The Mirror Has Two Faces'' and ''A Beautiful Mind'', the latter of which is based on the biography of the mathematician and Nobel laureate John Forbes Nash by Sylvia Nasar.[ The music of primes], Marcus du Sautoy's selection of films featuring prime numbers. Prime numbers are used as a metaphor for loneliness and isolation in the Paolo Giordano novel ''The Solitude of Prime Numbers'', in which they are portrayed as &quot;outsiders&quot; among integers.  