The Theory of Prime Number Classification

Prime numbers
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In the opinion of the 18th century British. The first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation. Obviously, in either of these equations, if we exchange A and B , we obtain another true statement.

Furthermore, it is true, but far less obvious, that this holds for every possible algebraic equation with rational coefficients relating the A and B values above in any such equation, swapping A and B yields another true equation. To prove this requires the theory of symmetric polynomials. We wish to describe the Galois group of this polynomial, again over the field of rational numbers. The polynomial has four roots. There are 24 possible ways to permute these four roots, but not all of these permutations are members of the Galois group.

The members of the Galois group must preserve any algebraic equation with rational coefficients involving A , B , C and D. The notion of a solvable group in group theory allows one to determine whether a polynomial is solvable in radicals, depending on whether its Galois group has the property of solvability. If all the factor groups in its composition series are cyclic, the Galois group is called solvable , and all of the elements of the corresponding field can be found by repeatedly taking roots, products, and sums of elements from the base field usually Q.

The other four roots are complex numbers. By the rational root theorem this has no rational zeros. Neither does it have linear factors modulo 2 or 3. Thus its Galois group modulo 3 contains an element of order 5.

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It is known that a Galois group modulo a prime is isomorphic to a subgroup of the Galois group over the rationals. This is one of the simplest examples of a non-solvable quintic polynomial. Serge Lang has said that Emil Artin found this example. The 3-adic integers, with selected corresponding haracters on their Pontryagin dual group In mathematics the p -adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems.

The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, p -adic numbers have the interesting property that they are said to be close when their difference is divisible by a high power of p — the higher the power the closer they are. This property enables p -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.

Their influence now extends far beyond this. For example, the field of p -adic analysis essentially provides an alternative form of calculus. More formally, for a given prime p , the field Q p of p -adic numbers is a completion of the rational numbers. The field Q p is also given a topology derived from a metric, which is itself derived from an alternative valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in Q p.

This is what allows the development of calculus on Q p , and it is the interaction of this analytic and algebraic structure which gives the p - adic number systems their power and utility. When dealing with ordinary real numbers, if we take p to be a fixed prime number, then any positive integer can be written as a base p expansion in the form. A definite meaning is given to these sums based on Cauchy sequences, using the absolute value as metric. With p -adic numbers, on the other hand, we choose to extend the base p expansions in a different way. Because in the p -adic world high positive powers of p are small and high negative powers are large, we consider infinite sums of the form:.

With this approach we obtain the p -adic expansions of the p -adic numbers. The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers; this allows us to, for example, write 1 as 1. The definition of a Cauchy sequence relies on the metric chosen, though, so if we choose a different one, we can.

7.1 Geometry of numbers

The Theory of Prime Number Classification This is an expository work of mathematical research into the prime numbers based on pattern methodology and. The Theory of Prime Number Classification This is an expository work of mathematical research into the prime numbers based on pattern.

The usual metric which yields the real numbers is called the Euclidean metric. In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field or more generally any Dedekind domain can be described by a certain group known as an ideal class group or class group. If this group is finite as it is in the case of the ring of integers of a number field , then the order of the group is called the class number. The multiplicative theory of a Dedekind domain is intimately tied to the structure of its class group.

For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain. Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions.

Additive number theory is concerned with the additive structure of the integers, such as Goldbach's conjecture that every even number greater than 2 is the sum of two primes. Developments within analytic number theory are often refinements of earlier techniques, which reduce the error terms and widen their applicability. For example, the circle method of Hardy and Little wood was conceived as applying to power series near the unit circle in the complex plane, it is now thought of in terms of finite exponential sums that is, on the unit circle, but with the power series truncated.

The great theorems and results within analytic number theory tend not to be exact structural results about the integers, for which algebraic and geometrical tools are more appropriate. Instead, they give approximate bounds and estimates for various number theoretical functions, as the following examples illustrate. Euclid showed that there are an infinite number of primes but it is very difficult to find an efficient method for determining whether or not a number is prime, especially a large number.

A related but easier problem is to determine the asymptotic distribution of the prime numbers; that is, a rough description of how many primes are smaller than a given number. Gauss, amongst others, after computing a large list of primes, conjectured that the number of primes less than or equal to a large number N is close to the value of the integral In Bernhard Riemann used complex analysis and a special meromorphic function now known as the Riemann zeta function to derive an analytic expression for the number of primes less than or equal to a real number x.

Remarkably, the main term in Riemann's formula was exactly the above integral, lending substantial weight to Gauss's conjecture. Riemann found that the error terms in this expression, and hence the manner in which the primes are distributed, are closely related to the complex zeros of the zeta function. In particular, they proved that if then This remarkable result is what is now known as the Prime Number Theorem.

It is a central result in analytic number theory. In one of the first applications of analytic techniques to number theory, Dirichlet proved that any arithmetic progression with a and q co prime contains infinitely many primes. The general case was proved by Hilbert in , using algebraic techniques which gave no explicit bounds. An important breakthrough was the application of analytic tools to the problem by Hardy and Little wood. One of the most useful tools in multiplicative number theory are Dirichlet series, which are functions of a complex variable defined by an infinite series Depending on the choice of coefficients , this series may converge everywhere, nowhere, or on some half plane.

In many cases, even where the series does not converge everywhere, the holomorphic function it defines may be analytically continued to a meromorphic function on the entire complex plane. The utility of functions like this in multiplicative problems can be seen in the formal identity hence the coefficients of the product of two Dirichlet series are the multiplicative convolutions of the original coefficients.

Furthermore, techniques such as partial summation and Tauberian theorems can be used to get information about the coefficients from analytic information about the Dirichlet series. Thus a common method for estimating a multiplicative function is to express it as a Dirichlet series or a product of simpler Dirichlet series using convolution identities ,. His theories contributed to Riemannian geometry,algebraic geometry, complex manifolds,and mathematical physics.

He is best known for his work in analysis,for defining the Riemann integral using Riemann sums. In the field of number theory,Riemann only wrote one paper ,establishing the importance of the Riemann Zeta function and its relation to prime numbers. Values with arguments close to zero including positive reals on the real half-line are presented in red. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

This function, as a function of a real argument, was introduced and studied by Leonhard Euler in the first half of the eighteenth century without using complex analysis, which was not available at that time. Bernhard Riemann in his memoir "On the Number of Primes Less Than a Given Magnitude" published in extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers. The values of the Riemann zeta function at even positive integers were computed by Euler.

The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known. No such simple expression is known for odd positive integers.

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The values of the zeta function obtained from integral arguments are called zeta constants. The following are the most commonly used values of the Riemann zeta function. Then this is the harmonic series. But its principal value. The reciprocal of this sum answers the question: What is the probability that two numbers selected at random are relatively prime? This appears when integrating Planck's law to derive the Stefan—Boltzmann law in physics. The connection between the zeta function and prime numbers was discovered by Euler, who proved the identity. The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic.

The Euler product formula can be used to calculate the asymptotic probability that s randomly selected integers are set-wise co prime. There are various expressions for the zeta-function as a Mellin transform.

If the real part of s is greater than one, we have. We can also find expressions which relate to prime numbers and the prime number theorem. These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Another series development using the rising factorial valid for the entire complex plane is. This can be used recursively to extend the Dirichlet series definition to all complex numbers. In number theory ,an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers.

The name arose from the case of the Riemann Zeta function , Where such a product representation was proved by Leonhard Euler. The Euler product formula for the Riemann zeta function reads where the left hand side equals the Riemann zeta function: The method of Eratosthenes used to sieve out prime numbers is employed in this proof. This sketch of a proof only makes use of simple algebra commonly taught in high school.

This was originally the method by which Euler discovered the formula. There is a certain sieving property that we can use to our advantage: Subtracting the second from the first we remove all elements that have a factor of 2: Repeating for the next term: Subtracting again we get: It can be seen that the right side is being sieved. Repeating infinitely we get: This can be written more concisely as an infinite product over all primes p: This proves that there are infinitely many prime numbers. Considerable effort has been expended in primality- testing and integer factorization routines, for example -- procedures which are in principle trivial, but whose naive solution is untenable in large cases.

This field also considers integer quantities e. To help measure the sizes of their fields, the Egyptians invented geometry To help predict the positions of the planets, the Greeks invented trigonometry. Algebra was invented to deal with equations that arose when mathematics was used to model the world. The list goes on, and it is not just historical.

If anything, computation is more important than ever. Much of modern technology rests on algorithms that compute quickly: In pure mathematics we also compute, and many of our great theorems and conjectures are, at root, motivated by computational experience. It is said that Gauss , who was an excellent computationalist, needed only to work out a concrete example or two to discover, and then prove, the underlying theorem.

While some branches of pure mathematics have perhaps lost contact with their computational origins, the advent of cheap computational power and convenient mathematical software has helped to reverse this trend. One mathematical area where the new emphasis on computation can be clearly felt is number theory, and that is the main topic of this article.

A prescient call to arms was issued by Gauss as long ago as The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors, is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length.

Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers.

Further, the dignity of the science itself seems to require that every possible means be explored for solution of a problem so elegant and so celebrated. Factorization into primes is a very basic issue in number theory, but essentially all branches of number theory have a computational component. And in some areas there is such a robust computational literature that we discuss the algorithms involved as mathematically interesting objects in their own right.

When the numbers are very large, no efficient, non-quantum integer factorization algorithm is known; an effort concluded in by several researchers factored a digit number RSA- , utilizing hundreds of machines over a span of 2 years The presumed difficulty of this problem is at the heart of widely used algorithms in cryptography such as RSA. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. Not all numbers of a given length are equally hard to factor.

The hardest instances of these problems for currently known techniques are semi primes, the product of two prime numbers. When they are both large, for instance more than bits long, randomly chosen, and about the same size but not too close, e. An algorithm that efficiently factors an arbitrary integer would render RSA-based public-key cryptography insecure.

This image demonstrates the prime decomposition of A shorthand way of writing the resulting prime factors is By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization.

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A special case for 1 is not needed using an appropriate notion of the empty product. However, the fundamental theorem of arithmetic gives no insight into how to obtain an integer's prime factorization; it only guarantees its existence. Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm. However, this is not the case with a special-purpose factorization algorithm, since it may not apply to the smaller factors that occur during decomposition, or may execute very slowly on these values.

The most difficult integers to factor in practice using existing algorithms are those that are products of two large primes of similar size, and for this reason these are the integers used in cryptographic applications. The largest such semi prime yet factored was RSA, a bit number with decimal digits, on December 12, This factorization was a collaboration of several research institutions, spanning two years and taking the equivalent of almost years of computing on a single-core 2.

Like all recent factorization records, this factorization was completed with a highly optimized implementation of the general number field sieve run on hundreds of machines. If a large, b -bit number is the product of two primes that are roughly the same size, then no algorithm has been published that can factor in polynomial time, i.

The best published asymptotic running time is for the general number field sieve GNFS algorithm, which, for a b -bit number n, is: For an ordinary computer, GNFS is the best published algorithm for large n more than about digits. For a quantum computer, however, Peter Shor discovered an algorithm in that solves it in polynomial time. This will have significant implications for cryptography if a large quantum computer is ever built. Shor's algorithm takes only O b 3 time and O b space on b -bit number inputs.

In , the first seven-qubit quantum computer became the first to run Shor's algorithm. It factored the number When discussing what complexity classes the integer factorization problem falls into, it's necessary to distinguish two slightly different versions of the problem:. This is the version solved by most practical implementations. This version is useful because most well-studied complexity classes are defined as classes of decision problems, not function problems.

This is a natural decision version of the problem, analogous to those frequently used for optimization problems, because it can be combined with binary search to solve the function problem version in a logarithmic number of queries. It is not known exactly which complexity classes contain the decision version of the integer factorization problem. It is known to be in both NP and co-NP. This is because both YES and NO answers can be verified in polynomial time given the prime factors we can verify their primality using the AKS primality test, and that their product is N by multiplication.

The fundamental theorem of arithmetic guarantees that there is only one possible string that will be accepted providing the factors are required to be listed in order , which shows that the problem is in both UP and co-UP. It is known to be in BQP because of Shor's algorithm. It is suspected to be outside of all three of the complexity classes P, NP-complete, and co-NP- complete. It is therefore a candidate for the NP-intermediate complexity class.

That would be a very surprising result, and therefore integer factorization is widely suspected to be outside both of those classes. Many people have tried to find classical polynomial-time algorithms for it and failed, and therefore it is widely suspected to be outside P. In contrast, the decision problem "is N a composite number?

Specifically, the former can be solved in polynomial time in the number n of digits of N with the AKS primality test. In addition, there are a number of probabilistic algorithms that can test primality very quickly in practice if one is willing to accept the vanishingly small possibility of error. The ease of primality testing is a crucial part of the RSA algorithm, as it is necessary to find large prime numbers to start with.

The classical Geometry of Numbers due to Minkowski begins with statements of Euclidean geometry on lattices A convex body contains a lattice point if its volume is large enough ; by extension this becomes the study of quadratic forms on lattices, and thus a method of investigating regular packings of spheres, say. But one may also investigate algebraic geometry with number theory, that is, one may study varieties such as algebraic curves and surfaces and ask if they have rational or integral solutions points with rational or integral coordinates.

Number theory

This topic includes the highly successful theory of elliptic curves where the rational points form a finitely generated group and finiteness results e. Siegel's, Thue's, or Faltings's which apply to integral or higher-genus situations. In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by Hermann Minkowski Minkowski's theorem on successive minima, sometimes called Minkowski's second theorem, is a strengthening of his first theorem and states that [] Later research in the geometry of numbers In research on the geometry of numbers was conducted by many number theorists including Louis Mordell, Harold Davenport and Carl Ludwig Siegel.

In recent years, Lenstra, Brion, and Barvinok have developed combinatorial theories that enumerate the lattice points in some convex bodies. Influence on functional analysis. Minkowski's geometry of numbers had a profound influence on functional analysis. Minkowski proved that symmetric convex bodies induce norms in finite-dimensional vector spaces.

Minkowski's theorem was generalized to topological vector spaces by Kolmogorov, whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a Banach space. Researchers continue to study generalizations to star-shaped sets and other non-convex sets. In Frank Morley, a professor at Haverford, discovered the following remarkable theorem.

The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle. If any hexagon convex or not is inscribed in a conic section and opposite sides are extended until they meet, then the three points of intersection will be collinear. The line is now called the Pascal line. When you try the applet, do not forget to try the non convex configurations! In fact, given a hexagon, we could keep the vertices fixed and permute their order to obtain other hexagons. A little combinatorics shows that there are 60 different hexagons for each collection of six points.

Each configuration has its own Pascal line. There is a lot known about these Pascal lines and their intersections. This last theorem is remarkable, not for what it says, but because of the difficulty of the proof. Lehmus asked for a purely geometric proof of the following elementary-looking theorem.

Any triangle with two angle bisectors of equal lengths is isosceles. For example, suppose we have the triangle shown below with angle bisectors and of the same length. Prove that and are the same length. Now there are many geometric and trigonometric proofs, but they are all tricky and are all proofs by contradiction. In Sylvester asked whether there exists a direct proof of this theorem. It appears that this is still an open problem.

From what I understand, there have been direct proofs,. By the Chinese remainder theorem, however, these calculations can be done in the isomorphic ring instead. Since p and q are normally of about the same size, that is about , calculations in the latter representation are much faster. Note that RSA algorithm implementations using this isomorphism are more susceptible to fault injection attacks. Let r complex points "interpolation nodes" be given, together with the complex data , for all and.

The general Hermite interpolation problem asks for a polynomial taking the prescribed derivatives in each node: Introducing the polynomials the problem may be equivalently reformulated as a system of simultaneous congruences: By the Chinese remainder theorem in the principal ideal domain , there is a unique such polynomial with degree. A direct construction, in analogy with the above proof for the integer number case, can be performed as follows.

Define the polynomials and. The partial fraction decomposition of gives r polynomials with degrees such that so that. Then a solution of the simultaneous congruence system is given by the polynomial. Secret Sharing using the Chinese Remainder Theorem uses, along with the Chinese remainder theorem, special sequences of integers that guarantee the impossibility of recovering the secret from a set of shares with less than a certain cardinality.

The Prime-factor FFT algorithm contains an implementation. Proof using the Chinese Remainder Theorem: First, assume that k is a field otherwise, replace the integral domain k by its quotient field, and nothing will change. Then, the condition yields by linearity. Now, we notice that if are two elements of the index set I , then the two k -linear maps and are not proportional to each other because if they were, then and would also be proportional to each other, and thus equal to each other since since and are monoid homomorphism , contradicting the assumption that they be distinct.

Hence, their kernels and are distinct. Now, is a maximal ideal of for every since is a field , and the ideals and are co prime whenever since they are distinct and maximal. The Chinese Remainder Theorem for general rings thus yields that the map given by for all is an isomorphism, where. Under the isomorphism , this map corresponds to the map given by for every Now, yields for every vector in the image of the map.

Since is surjective, this means that for every vector. For any prime p we have. Where the coefficient is the kth elementary symmetric function of the roots, that is, the sum of the products of the numbers 1, 2…. Each of the numbers … is divisible by p. And reducing the equation mod we find, since p 5,p 0 mod And hence 0 mod ,as required. Determine whether is a quadratic residue or nonresidue mod solution. It is about constructing and analyzing protocols that overcome the influence of adversaries and which are related to various aspects in information security such as data integrity, authentication and non-repudiation.

More generally, it is about constructing and analyzing protocols that overcome the influence of adversaries and which are related to various. Modern cryptography intersects the disciplines of mathematics, computer science, and electrical engineering.

Applications of cryptography include ATM cards, computer passwords, and electronic commerce. Modern cryptography is heavily based on mathematical theory and computer science practice; cryptographic algorithms are designed around computational hardness assumptions, making such algorithms hard to break in practice by any adversary. It is theoretically possible to break such a system but it is infeasible to do so by any known practical means.

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These schemes are therefore termed computationally secure; theoretical advances e. There exist information-theoretically secure schemes that provably cannot be broken even with unlimited computing power—an example is the one-time pad—but these schemes are more difficult to implement than the best theoretically breakable but computationally secure mechanisms. Before the modern era, cryptography was concerned solely with message confidentiality i. Encryption was used to attempt to ensure secrecy in communications, such as those of spies, military leaders, and diplomats.

Reconstructed ancient Greek scytale rhymes with "Italy" , an early cipher device.

Relation to ideals and arithmetic geometry

The most difficult integers to factor in practice using existing algorithms are those that are products of two large primes of similar size, and for this reason these are the integers used in cryptographic applications. An early example, from Herodotus, concealed a message—a tattoo on a slave's shaved head—under the regrown hair Another Greek method was developed by Polybius now called the "Polybius Square". The partial fraction decomposition of gives r polynomials with degrees such that so that. There are 24 possible ways to permute these four roots, but not all of these permutations are members of the Galois group. Number theory is an art enjoyable and pleasing to everybody. Triangular numbers, square numbers, pentagonal numbers, etc. That internal state is initially set up using the secret key material.

The earliest forms of secret writing required little more than local pen and paper analogs, as most people could not read. More literacy, or literate opponents, required actual cryptography. The main classical cipher types are transposition ciphers, which rearrange the order of letters in a message e. Simple versions of either have never offered much confidentiality from enterprising opponents.

An early substitution cipher was the Caesar cipher, in which each letter in the plaintext was replaced by a letter some fixed number of positions further down the alphabet. Suetonius reports that Julius Caesar used it with a shift of three to communicate with his generals. Atbash is an example of an early Hebrew cipher. The earliest known use of cryptography is some carved ciphertext on stone in Egypt ca BCE , but this may have been done for the amusement of literate observers rather than as a way of concealing information. Cryptography is recommended in the Kama Sutra ca BCE as a way for lovers to communicate without inconvenient discovery.

The Greeks of Classical times are said to have known of ciphers e. An early example, from Herodotus, concealed a message—a tattoo on a slave's shaved head—under the regrown hair Another Greek method was developed by Polybius now called the "Polybius Square". More modern examples of steganography include the use of invisible ink, microdots, and digital watermarks to conceal information. Essentially all ciphers remained vulnerable to cryptanalysis using the frequency analysis technique until the development of the polyalphabetic cipher, most clearly by Leon Battista Alberti around the year , though there is some indication that it was already known to Al- Kindi.

Alberti's innovation was to use different ciphers i. He also invented what was probably the first automatic cipher device, a wheel which implemented a partial realization of his invention. Although frequency analysis is a powerful and general technique against many ciphers, encryption has still often been effective in practice, as many a would-be cryptanalyst was unaware of the technique.

Breaking a message without using frequency analysis essentially required knowledge of the cipher used and perhaps of the key involved, thus making espionage, bribery, burglary, defection, etc. Just as the development of digital computers and electronics helped in cryptanalysis, it made possible much more complex ciphers. Furthermore, computers allowed for the encryption of any kind of data representable in any binary format, unlike classical ciphers which only encrypted written language texts; this was new and significant. Computer use has thus supplanted linguistic cryptography, both for cipher design and cryptanalysis.

Many computer ciphers can be characterized by their operation on binary bit sequences sometimes in groups or blocks , unlike classical and mechanical schemes, which generally manipulate traditional characters i. However, computers have also assisted cryptanalysis, which has compensated to some extent for increased cipher complexity.

Nonetheless, good modern ciphers have stayed ahead of cryptanalysis; it is typically the case that use of a quality cipher is very efficient i. Credit card with smart-card capabilities. The 3-bymm chip embedded in the card is shown, enlarged. Smart cards combine low cost and portability with the power to compute cryptographic algorithms.

Symmetric-key cryptography refers to encryption methods in which both the sender and receiver share the same key or, less commonly, in which their keys are different, but related in an easily computable way. This was the only kind of encryption publicly known until June A block cipher enciphers input in blocks of plaintext as opposed to individual characters, the input form used by a stream cipher.

Despite its deprecation as an official standard, DES especially its still-approved and much more secure triple-DES variant remains quite popular; it is used across a wide range of applications, from ATM encryptionto e-mail privacyand secure remote access. The fundamental theorem of arithmetic asserts that every nonzero integer can be written as a product of primes in a unique way, up to ordering and multiplication by units.

There are other ways to prove this fact, but Euclid's way is still considered the most elegant. A prime gap of 1 happens only once, i. It is conjectured that all even prime gaps happen infinitely often. A prime gap of 2 between the twin primes is conjectured to happen infinitely often, this being the twin primes conjecture.

This is the latest approved revision, approved on 16 September The draft has 34 changes awaiting review. Retrieved from " http: Articles needing more work Prime numbers.

4.7 Galois theory

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