Prime number

A prime number is a Entier naturalness, admitting two exactly dividing distinct : 1 and itself. By opposition, the others are known as “made up” (implied composed like product of two integers different from 1). For example 12 = 2×6 is composed, just like 21 = 3×7 or 7×3, but 11 is first because 1 and 11 is the only dividers of 11. The prime numbers lower than 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 and 97.

Such lists can be obtained thanks to various methods of calculating.

The concept of prime number is a basic concept in elementary Arithmétique: the fundamental Théorème of arithmetic the ensures that a made up number is factorisable in a product of prime numbers, and this factorization is single near with the order of the factors. She admits important generalizations in branches of more advanced mathematics, for example the Algebraic theory of the numbers, which thus take in their turn the name of arithmetic. In addition, of many industrial applications of arithmetic on algorithmic knowledge prime numbers rest, and sometimes more precisely on the difficulty of the algorithmic problems which are dependant for them; for example certain cryptographic systems and of the methods of transmission of information. The prime numbers are also used to build tables of chopping and to constitute generating of pseudo-random numbers.

Historical elements

Notches found on the Os of Ishango gone back to -23  000 years front J.C., updated by the archeologist Jean de Heinzelin de Braucourt and former to the appearance of the writing (towards -3  500 years before JC.), seem to isolate four prime numbers 11,13,17 and 19. Certain archeologists interpret it like the proof of the knowledge of the prime numbers. However, there exists discoveries not enough making it possible to determine real knowledge of this old period.

Dried clay shelves allotted to civilizations which followed one another in Mésopotamie in Euphrasie during front IIemillénaire J.C show the solution to problem arithmetic and attest first knowledge of the time. Calculations required to know opposite tables of entireties (reciprocal ) of which some were found. In the sexagesimal System used by the Babylonian civilization to write the entireties, reciprocal dividers of the powers of 60 ( regular numbers ) are calculated easily: for example, to divide by 24, it is to multiply by $2.60+30$ and to shift of two places the row. Their knowledge required a good comprehension of the multiplication, division and factorization of entireties.

In Egyptian mathematics , fractional calculation required knowledge on the operations, divisions of entireties and factorizations. The Egyptians noted only the opposite of entireties (1/2, 1/3, 1/4, 1/5,…) ; the writing of the fractions was done while adding with the opposite of entireties, if possible without repetition (1/2+1/6 instead of 1/3+1/3). To have a list of the first prime numbers was to be necessary.

The first undeniable trace of the presentation of the prime numbers goes back to the Antiquité (towards -300 front J.C.), and is in the Éléments of Euclide (volumes with). Euclide gives the definition of the prime numbers, the proof of their infinitude, the definition of the highest common factor (pgcd) and of the lowest common multiple (ppcm), and the algorithms to determine them, called today algorithms of Euclide. Knowledge presented is however quite former for him.

Structures algebraic, topological, and prime numbers

The concept of prime number is related to the study of the multiplicative structure of the ring of the relative entireties. The fundamental theorem of arithmetic, based on the Lemma of Euclide, elucidates this structure by ensuring that entire factorizes itself in a product of prime numbers, in a single way the order of the factors near. This theorem makes it possible to determine concepts of Pgcd, Ppcm, and of Prime numbers between them, which are useful for the resolution of some equations diophantiennes, in particular the characterization of the triplets pythagoricians.

Other natural problems are considered, like the determination of the proportion of entireties first to a fixed entirety. The introduction of more advanced algebraic structures makes it possible to quickly solve this problem within the framework of the modular Arithmétique. Many traditional theorems of arithmetic nature can be stated, like the Petit theorem of Fermat, or the Théorème of Wilson; or of the theorems of algebraic nature like the Theorem of the Chinese remainders.

The theorem of the Chinese remainders is a first result in the study of the finished abelian groups. It is in fact sufficient to entirely describe the structure of these groups, which is thus partly related to the decomposition in product of factors first their cardinals. The things are more complicated for the nonabelian groups, however, the study again bases on the decomposition in factors first their cardinals, through the Théorie of Sylow.

The prime numbers also intervene in the topological structures . The body of the rational numbers admits a usual topological structure, which gives by completion the body of the real numbers. For each prime number p , another structure topological can be built, on the standard basis following: if $x=\; \backslash \; frac\; \{has\}\; \{B\}$ not no one in irreducible form is a rational number and that $p^\; \backslash \; alpha$ and $p^\; \backslash \; beta$ is more the great powers of p dividing has and B , the p-adic Norme of X is $p^\; \{\backslash \; beta\; \backslash \; alpha\}$. By supplementing the body of rational the following this standard, one obtains the body of the numbers p-adic, introduced by Kurt Hensel at the beginning of the 20th century. The Théorème of Ostrowski ensures that these p-adic standards and the usual standard are only on the body of the rational numbers, except for equivalence.

Prime numbers private individuals

Prime numbers and numbers of Fermat

Prime numbers of the form:

$F\_n\; =\; 2^\; \{2^n\}\; +\; 1$

the prime numbers of Fermat are called. Pierre de Fermat had conjectured that all these numbers were to be first. However, the only known prime numbers of Fermat are:

• $F\_0\; =\; 2^\; \{1\}\; +\; 1\; =\; 3\; \backslash ,$
• $F\_1\; =\; 2^\; \{2\}\; +\; 1\; =\; 5\; \backslash ,$
• $F\_2\; =\; 2^\; \{4\}\; +\; 1\; =\; 17\; \backslash ,$
• $F\_3\; =\; 2^\; \{8\}\; +\; 1\; =\; 257\; \backslash ,$
• $F\_4\; =\; 2^\; \{16\}\; +\; 1\; =\; 65\; \backslash ,\; 537\; \backslash ,$

The number of Fermat $F\_5$ is not first: it is divisible by 641. It is about the first counterexample to this conjecture of Fermat, discovered by Euler in 1732.

The greatest known prime number and prime numbers of Mersenne

The prime numbers of the form 2^p-1, where p is itself a prime number, are called prime numbers of Mersenne. There exists an effective test to determine if a number of this form is or not first: the Test of primality of Lucas-Lehmer. The great prime numbers thus are often sought in this form, and largest Nombre known first is 2^{: 32582657}-1, it comprises: 9808358 digits in decimal writing. It is about the 44 {{E}} prime number of known Mersenne ( M_{: 32582657} ) and was announced the September 4th 2006 thanks to the efforts of a collaboration which bears the name of GIMPS.

The Electronic Frontier Foundation offers a price of co-operative calculation of an amount of: 100000 USD for the discovery of a prime number of at least ten million decimal digits, in order to encourage the Net surfers to contribute to the scientists solution to problem by the Calculation distributed.

Great illegal prime numbers

Some of the greatest prime numbers not having a particular form (i.e. not being able to be written using a simple formula as prime numbers of Mersenne) were found by taking a piece of pseudo-random binary data, and by converting them into a n number, while multiplying by 256^k where k is a certain strictly positive entirety, and by seeking prime numbers possible in the interval + 1,256^k (n + 1) - 1.

To launch an advertizing blow against the act of copyright DIGITAL Millennium and the other implementations of the Treated copyright WIPO, some people applied this method to various varied forms of the code DeCSS, by creating the whole of the illegal prime numbers. Such numbers, when they are converted into binary and are carried out in a Computer program, enfreignent the law in force in one or more jurisdictions of the United States of America.

Calculation of the prime numbers

Sift of Eratosthène and algorithm by tests of division

The first Algorithm S to decide if a number is first consist in trying to divide it by all the numbers lower than its square root: if it is divisible by one of them, it is composed, and if not, it is first. However, the algorithm deduced from this formulation can be made more effective: it suggests many divisions useless, for example, if a number is not divisible by 2 , it is useless to test if it is divisible by 4 . In fact, it is enough to test its divisibility by all the prime numbers lower than his square root.

The Crible of Ératosthène is a method resting on this idea; it provides in fact the list of all the prime numbers lower than a fixed value N :

• One forms the list of the entireties of 2 with N ,
• One takes the first number of this list (not yet barred), that is to say initially 2,
• One bars all the multiple entireties of the number selected, while starting with its square (since 2*i, 3*i,… (i-1) *i were already barred as multiples of 2,3,…)
• One repeats this last operation by considering the next number not barred.
• As soon as one is to be sought the multiples of the numbers exceeding the square root of N , one finishes the algorithm.

The numbers which remain not barred at the end of the process are the prime numbers lower than N . This algorithm is of algorithmic Complexité exponential.

The screen of Ératosthène thus provides more information than the only primality of N . If only this information is wished, an alternative sometimes more effective consists in testing the divisibility of N only by small numbers first in a list fixed as a preliminary (for example 2,3 and 5), then by all the integers lower than the square root of N which are not divisible per any the small numbers first selected; that brings to test divisibility by numbers not first (for example 49 if the small first are 2,3 and 5 and that N exceeds 2500), but a choice of a sufficient number of small numbers first must make it possible to control the number of useless tests carried out.

Other algorithms

An alternative of the screen of Ératosthène is the Crible of Sundaram which consists in forming the products of odd numbers. The numbers which are not reached by this métode are the prime numbers odd, i.e. all the prime numbers except 2 . In addition, starting from the screen of Ératosthène, the factorization of the entirety N can easily be found. Other more general methods relating to this problem more difficult than simply to determine the primality are also called methods of the screen, most effective being currently the general screen of the bodies of numbers.

The algorithms presented previously have a too important complexity to be able to be carried out in the long term, even with the most powerful computers, when N becomes large.

Another class of algorithm consists in testing the entirety N for a family of properties checked by the prime numbers: if it a property of this family is not checked for N , then it is made up; on the other hand, the fact that one of the properties of the family is checked for N is not enough to ensure the primality. However, if this family is such as a made up number does not check at least the half of the properties concerned, then a number N which checks K properties of the family will have a probability higher than 1-2^-k to be first: it is probably stated first starting from a value of K to be chosen by the user; a number probably declared first, but which is not first is called Nombre pseudo-first. A test based on this principle is called probabilistic test of primality. Such tests often rest on the Petit theorem of Fermat, bringing to the Test of primality of Fermat, and to its refinements: the Test of primality of Solovay-Strassen and that of Miller-Rabin, which is improvements, because they admit less pseudo-first numbers.

The algorithm AKS developped at the point in 2002 makes it possible to determine if a number given NR is first by using a polynomial computing time.

Formulas leading to the prime numbers

Many formulas were sought to generate the prime numbers. More the high level of requirement would be to find a formula which with an entirety N associates N E prime number. In a way a little more flexible, one can be satisfied to require a function F which with entire N associates a prime number and such as each value taken is it only once. Lastly, it is wished that the function be calculable in practice. For example, the theorem of Wilson ensures:

If $p$ is a prime number then $(p-1)!\; \backslash \; equiv\; -1\; \backslash \; MOD\; p$

and the function:

$f\; \backslash \; left\; (N\; \backslash \; right)\; =\; 2\; +\; \backslash \; left\; (\{2\; \backslash \; left\; (N!\; \backslash \; right)\}\; \backslash \; MOD\; \{\backslash \; left\; (n+1\; \backslash \; right)\}\; \backslash \; right)\; \backslash ,$

give all the prime numbers, only the prime numbers, and only the value 2 is taken several times. However, the calculation of the factorial is crippling.

The search for such functions was in particular undertaken among the functions polynomials, undertake with the negative result that a polynomial with complex coefficients , even with several unspecified, whose values with the natural entireties have as an absolute value of the prime numbers, is a constant polynomial. The search for polynomials checking a weaker property developed starting from the concept of Ensemble diophantien of integers; such units can be characterized like the whole of values taken by polynomials (with several variables) with whole coefficients in the strictly positive entireties. A work undertaken in the years 1960 and 1970, in particular by Putnam, Matijasevic, Davis, Robinson, makes it possible to show that the whole of the prime numbers is diophantien, leading to the existence of polynomial which take with the positive whole values variables like values the prime numbers. The writing of various explicit polynomials was then possible, with various numbers of variables, and various degrees. In particular, the following polynomial, of degree 25 to 26 variables (of with Z has), was determined by Jones, Sato, Wada and Wiens in 1976:

(1

− '' W ''. '' Z '' + '' H '' + '' J '' − '' Q '' 2

− 2. '' N '' + '' p '' + '' Q '' + '' Z '' − '' E '' 2

− '' has '' {{exp|2}}. '' there '' {{exp|2}} − '' there '' {{exp|2}} + 1 − '' X '' {{exp|2}} 2

− '' E '' {{exp|3}}. ('' E '' + 2). ('' has '' + 1) {{exp|2}} + 1 − '' O '' {{exp|2}} 2

− 16. ('' K '' + 1) {{exp|3}}. ('' K '' + 2). ('' N '' + 1) {{exp|2}} + 1 − '' F '' {{exp|2}} 2

− (('' has '' + '' U '' {{exp|2}}. ('' U '' {{exp|2}} − '' has '')){{exp|2}} − 1). ('' N '' + 4. '' D ''. '' there '') {{exp|2}} + 1 − ('' X '' + '' C ''. '' U '') {{exp|2}} 2

− '' has ''. '' I '' + '' K '' + 1 − '' L '' − '' I '' 2

− ('' G ''. '' K '' + 2. '' G '' + '' K '' + 1). ('' H '' + '' J '') + '' H '' − '' Z '' 2

− 16. '' R '' {{exp|2}}. '' there '' {{exp|4}}. ('' has '' {{exp|2}} − 1) + 1 − '' U '' {{exp|2}} 2

− '' p '' − '' m '' + '' L ''. ('' has '' − '' N '' − 1) + '' B ''. (2. '' has ''. '' N '' + 2. '' has '' − '' N '' {{exp|2}} − 2. '' N '' − 2) 2

− '' Z '' − '' p ''. '' m '' + '' p ''. '' L ''. '' has '' − '' p '' {{exp|2}} '' L '' + '' T ''. (2. '' has ''. '' p '' − '' p '' {{exp|2}} − 1) 2

− '' Q '' − '' X '' + '' there ''. ('' has '' − '' p '' − 1) + '' S ''. (2. '' has ''. '' p '' + 2. '' has '' − '' p '' {{exp|2}} − 2. '' p '' − 2) 2

− '' has '' {{exp|2}}. '' L '' {{exp|2}} − '' L '' {{exp|2}} + 1 − '' m '' {{exp|2}} 2

− '' N '' + '' L '' + '' v '' − '' there '' 2

). ( K + 2)

The overall concept diophantien more generally developed starting from the problems arising from the tenth problem of Hilbert on the equations diophantiennes.

Distribution of the prime numbers

Infinity of the prime numbers

Euclide showed in its Éléments (proposal 20 of the book) that the prime numbers are in greater quantity than any quantity suggested of prime numbers . In other words, there exists a Infinité of prime numbers. The demonstration of Euclide rests on the observation that a finished family p₁,…, p_n of prime numbers being given, any prime number dividing the product of the elements of this family increased by 1 is apart from this family (and such a divider exists, which is also proven by Euclide).

Others Démonstration S of the infinity of the prime numbers were given. The proof of Euler uses the fact that $\backslash \; prod\_\; \{p\; \backslash ;\; \{\backslash \; rm\; first\}\}\; \{1\; \backslash \; over\; 1\; -\; \{1\; \backslash \; over\; p\}\}\; =\; \backslash \; sum\_1^\; \{\backslash \; infty\}\; \{1\; \backslash \; over\; N\}$ which is a divergent series. The product must thus comprise an infinity of factors. Furstenberg provides a proof using a topological argumentation .

Projections of the 19th century

The result on the infinity of the prime numbers pleasing of the more precise questions concerning the function which with a Real number X associates $\backslash \; pi\; (X)$, the number of prime numbers lower than X , and which thus tends towards the infinite one. An important conjecture at the 19th century, formulated by Adrien-Marie Legendre and Carl Friedrich Gauss, was that this Fonction of account of the prime numbers is equivalent to the function $\backslash \; frac\; \{X\}\; \{\backslash \; ln\; (X)\}$ when X tends towards the infinite one, i.e. the proportion of prime numbers among the numbers lower than X (either $\backslash \; frac\; \{\backslash \; pi\; (X)\}\{X\}$) tightens towards 0 at the speed of $\backslash \; frac\; \{1\}\; \{\backslash \; ln\; (X)\}$. Before the demonstration of the conjecture at the end of the century, a partial result had been shown by Pafnouti Tchebychev, the existence of two explicit constants C and D such as one has the framing, for X rather large:

$C\; \backslash \; Leq\; \backslash \; pi\; (X)\; \backslash \; frac\; \{\backslash \; ln\; (X)\}\{X\}\; \backslash \; Leq\; D.$

The inequality of Tchebychef in particular made it possible to show the Postulat of Bertrand according to whom in any interval of natural entireties between an entirety and its double exists at least a prime number. More generally, the results on the function of account of the prime numbers make it possible to obtain results on the prime number; for example the results of Ishikawa of 1934 are direct consequences of the theorems of Tchebychev: p_n + p_{n + 1} > p_{n + 2} and p_np_m > p_{n + m}, where p_n indicates to it prime number (and thus p₁=2); or, according to a result of Felgner of 1990: 0,91 N ln (N) < p_n < 1,7 N ln (N) , where p_n.

The analytical demonstration of Euler on the infinity of the prime numbers can be seen like a first step towards the solution to problem more advanced. It primarily consists in studying the behavior of the Fonction zeta of Riemann into 1 by means of what it is agreed to call a Produit eulérien, and to deduce the divergence from it from the series of the opposite of the prime numbers. By resuming this study, by means of a tool called Character of Dirichlet, and by using in the place of the function zeta of Riemann of the called similar functions function L of Dirichlet, Dirichlet was able to adapt the demonstration to the prime numbers in arithmetic progressions: if has and B is first between them, then there exists an infinity of prime numbers form aq+b . More precisely, the prime numbers are équirépartis between the various arithmetic progressions of reason has (i.e. with has fixed, and B variable among the various invertible remainders in Euclidean division by has ).

The conjecture of Legendre and Gauss was shown independently by Jacques Hadamard and Charles-Jean de la Vallee poussin in 1896, and bears the name of Théorème of the prime numbers. These demonstrations require powerful tools of Analyze complexes to show a statement of arithmetic and real analysis. A strategy for these demonstrations is the study of the function zeta of Riemann on a field larger than a simple vicinity of 1 : it is necessary to control it (i.e. to raise its module) in the vicinity of the vertical right-hand side of the numbers of real part 1 in the complex plan. In particular, the study of the function zeta of Riemann becomes a central theme in analytical Théorie of the numbers, in particular the Hypothèse of Riemann on the localization from its zeros, still not shown, which would have strong consequences on the study of the function of account of the prime numbers. Later on, of the demonstrations were proposed without recourse to the complex analysis (by Erdös and Selberg in the middle of the 20th century). However, the power of the tools for complex analysis led to the development of a whole branch of mathematics: the analytical theory of the numbers.

Theorem of Green and CAT

A theorem shown in 2004 per Benjamin Green and Terence CAT generalizes in particular the theorem of Dirichlet by ensuring that for entire K , there exists an infinity of continuations of K prime numbers which follow one another in an arithmetic progression, i.e. form: $a,\; a+b,\; a+2b,\; \backslash \; dowries,\; a+\; (k-1)\; b.$ The theorem of Green and CAT is in fact much more extremely than this statement alone: for example, they are able to affirm that such an arithmetic progression exists, with entireties all smaller than: $2^\{2^\{2^\{2^\{2^\{2^\{2^\{100k\}\}\}\}\}\}\}.$ Ilsl also ensure that for entire K and any strictly positive reality $\backslash \; delta$, for all X sufficiently large, if P is a whole of prime number lower than X containing at least $\backslash \; delta\; \backslash \; pi\; (X)$ elements, then P contains at least an arithmetic progression of prime numbers cash K terms.

A conjecture

Many results and conjectures about the distribution of the prime numbers are contained in the following general conjecture. Either f₁ ,…, f_k of the polynomials of degree 1, irreducible and checking the property that for any prime number p there is at least an entirety N among 0 ,…, p-1 such as p does not divide the product of the f_i (N) . One notes $\backslash \; Omega\; (p)$ the complementary one to p of the number of such entireties. Such a whole of polynomials is known as acceptable; one seeks to know the proportion of entireties in which the polynomials take simultaneously values first, and to limit itself to whole of acceptable polynomials makes it possible to avoid commonplace cases like f₁ (T) =t , and f₂ (T) =t+1 . It is then conjectured that the number of entireties N smaller than a reality X such as the values f₁ (N) ,…, f_k (N) are simultaneously first, is, for X rather large, about: $\backslash \; left\; (\backslash \; prod\_\; \{p\}\; \backslash \; frac\; \{1\; \backslash \; frac\; \{\backslash \; Omega\; (p)\}\{p\}\}\; \{\backslash \; left\; (1\; \backslash \; frac\; \{1\}\; \{p\}\; \backslash \; right)\; ^k\}\; \backslash \; right)\; \backslash \; frac\; \{X\}\; \{\backslash \; log|f\_1\; (X)|\backslash \; dowries\; \backslash \; log|f\_k\; (X)|\}.$ The theorem of the prime numbers corresponds to the case k=1 and f_t=t , the theorem of Dirichlet with k=1 and f_t=at+b , and for k=2 , f₁ (T) =t and f₂ (T) =t+2 , one obtains to a version quantitative (and thus more general) of the Conjecture of the prime numbers twins.

Generalizations of the prime numbers

Open-ended questions

There are many open-ended questions on the prime numbers. For example:

• the Conjecture of Goldbach: can any even number strictly higher than 2, be written like summons of two prime numbers?
• Conjecture of the prime numbers twins: a couple of Prime numbers twins is a pair of prime numbers of which the difference is equal to 2, like 11 and 13. Does there exist an infinity of twins first?
• All Suite of Fibonacci it contains an infinity of prime numbers?
• does there Exist an infinity of prime numbers of Fermat?
• Y does it have an infinity of prime numbers form N ² + 1?
• Y does it have an infinity of prime numbers factorial?
• Y does it have an infinity of prime numbers primoriels?
• Either the continuation, known as of Euclide-Mullin, first u₁=2 term and such as the u_n term or more the small number first divider of the product of the u_i terms, for i, increased 1 . Do all the prime numbers appear in this continuation? It is a conjecture of Daniel Shanks.

Quotations

• “a prime number is a number which does not break when one drops it by ground. ” Paul Erdős

• “the mathematicians tried in vain to up to now discover some order in the progression of the prime numbers, and one takes place to believe that it is a mystery to which the human spirit will be able to never penetrate. To be convinced of it, one has only to throw the eyes on the tables of the prime numbers that some tried hard to continue beyond a hundred and thousand and one will realize initially that it reigns there no order nor rule. ” Euler.

See too

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