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    Get your house holiday ready See more. Bargain tech accessories See more. Many mathematicians have worked on primality tests for numbers larger than those where trial division is practicably applicable.

    Most early Greeks did not even consider 1 to be a number, [34] [35] so they could not consider its primality. A few mathematicians from this time also considered the prime numbers to be a subdivision of the odd numbers, so they also did not consider 2 to be prime.

    However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number.

    If the definition of a prime number were changed to call 1 a prime, many statements involving prime numbers would need to be reworded in a more awkward way.

    For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with different numbers of copies of 1.

    Writing a number as a product of prime numbers is called a prime factorization of the number. The terms in the product are called prime factors.

    When a prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number: The central importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic.

    More strongly, this product is unique in the sense that any two prime factorizations of the same number will have the same numbers of copies of the same primes, although their ordering may differ.

    Primes can thus be considered the "basic building blocks" of the natural numbers. Some proofs of the uniqueness of prime factorizations are based on Euclid's lemma: There are infinitely many prime numbers.

    Another way of saying this is that the sequence. 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 , [47] Furstenberg's proof using general topology , [48] and Kummer's elegant proof.

    Euclid's proof [50] shows that every finite list of primes is incomplete. Because there is no finite list of all the primes, there must be infinitely many primes.

    The numbers formed by adding one to the products of the smallest primes are called Euclid numbers. There is no known efficient formula for primes.

    For example, there is no non-constant polynomial , even in several variables, that takes only prime values. One possible formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.

    This can be used to obtain a single formula with the property that all its positive values are prime.

    Other examples of prime-generating formulas come from Mills' theorem and a theorem of Wright. Many conjectures revolving about primes have been posed.

    Often having an elementary formulation, many of these conjectures have withstood proof for decades: Another type of problem concerns prime gaps , the differences between consecutive primes.

    The existence of arbitrarily large prime gaps can be seen by noting that the sequence n! It is conjectured that there are infinitely many twin primes , pairs of primes with difference 2; this is the twin prime conjecture.

    Their infinitude and density are the subject of the first Hardy—Littlewood conjecture , which can be motivated by the heuristic that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem.

    Analytic number theory studies number theory through the lens of continuous functions , limits , infinite series , and the related mathematics of the infinite and infinitesimal.

    This area of study began with Leonhard Euler and his first major result, the solution to the Basel problem. This function is closely connected to the prime numbers and to one of the most significant unsolved problems in mathematics, the Riemann hypothesis.

    Dirichlet's theorem on arithmetic progressions , in its basic form, asserts that linear polynomials. Although conjectures have been formulated about the proportions of primes in higher-degree polynomials, they remain unproven, and it is unknown whether there exists a quadratic polynomial that for integer arguments is prime infinitely often.

    Euler's proof that there are infinitely many primes considers the sums of reciprocals of primes,. The growth rate of this sum is described more precisely by Mertens' second theorem.

    In this sense, prime numbers occur more often than squares of natural numbers, although both sets are infinite.

    Because of Brun's theorem, it is not possible to use Euler's method to solve the twin prime conjecture , that there exist infinitely many twin primes.

    An arithmetic progression is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference.

    In an arithmetic progression, all the numbers have the same remainder when divided by the modulus; in this example, the remainder is 3.

    Because both the modulus 9 and the remainder 3 are multiples of 3, so is every element in the sequence. Therefore, this progression contains only one prime number, 3 itself.

    In general, the infinite progression. If they are relatively prime, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes.

    The Green—Tao theorem shows that there are arbitrarily long finite arithmetic progressions consisting only of primes. No quadratic polynomial has been proven to take infinitely many prime values.

    The Ulam spiral arranges the natural numbers in a two-dimensional grid, spiraling in concentric squares surrounding the origin with the prime numbers highlighted.

    Visually, the primes appear to cluster on certain diagonals and not others, suggesting that some quadratic polynomials take prime values more often than others.

    This function is an analytic function on the complex numbers. This equality between a sum and a product, discovered by Euler, is called an Euler product.

    In the terminology of abstract algebra , the ability to perform division means that modular arithmetic modulo a prime number forms a field or, more specifically, a finite field , while other moduli only give a ring but not a field.

    Several theorems about primes can be formulated using modular arithmetic. This picture of an order, absolute value, and complete field derived from them can be generalized to algebraic number fields and their valuations certain mappings from the multiplicative group of the field to a totally ordered additive group , also called orders , absolute values certain multiplicative mappings from the field to the real numbers, also called norms , [] and places extensions to complete fields in which the given field is a dense set , also called completions.

    The corresponding mapping to an additive group would be the logarithm of the absolute value, although this does not meet all the requirements of a valuation.

    A commutative ring is an algebraic structure where addition, subtraction and multiplication are defined. The integers are a ring, and the prime numbers in the integers have been generalized to rings in two different ways, prime elements and irreducible elements.

    An element is irreducible if it is neither a unit nor the product of two other non-unit elements. In the ring of integers, the prime and irreducible elements form the same set,.

    In an arbitrary ring, all prime elements are irreducible. The converse does not hold in general, but does hold for unique factorization domains.

    The fundamental theorem of arithmetic continues to hold by definition in unique factorization domains. Its prime elements are known as Gaussian primes.

    Rational primes the prime elements in the integers congruent to 3 mod 4 are Gaussian primes, but rational primes congruent to 1 mod 4 are not.

    Not every ring is a unique factorization domain. In order to extend unique factorization to a larger class of rings, the notion of a number can be replaced with that of an ideal , a subset of the elements of a ring that contains all sums of pairs of its elements, and all products of its elements with ring elements.

    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 every ideal in a Noetherian commutative ring as an intersection of primary ideals , which are the appropriate generalizations of prime powers.

    The spectrum of a ring is a geometric space whose points are the prime ideals of the ring. 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.

    These concepts can even assist with 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 that concerns the existence of square roots modulo integer prime numbers.

    By Lagrange's theorem , any group of prime order is a cyclic group , and by the Burnside theorem any group whose order is divisible by only two primes is solvable.

    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 mathematics [b] with the exception of use of prime numbered gear teeth to distribute wear evenly.

    Hardy prided themselves on doing work that had absolutely no military significance. This vision of the purity of number theory was shattered in the s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms.

    The most basic primality testing routine, trial division, is too slow to be useful for large numbers.

    One group of modern primality tests is applicable to arbitrary numbers, while more efficient tests are available for numbers of special types.

    Most primality tests only tell whether their argument is prime or not. Routines that also provide a prime factor of composite arguments or all of its prime factors are called factorization algorithms.

    Prime numbers are also used in computing for checksums , hash tables , and pseudorandom number generators. Another optimization is to check only primes as factors in this range.

    Each division produces a nonzero remainder, so 37 is indeed prime. Although this method is simple to describe, it is impractical for testing the primality of large integers, because the number of tests that it performs grows exponentially as a function of the number of digits of these integers.

    Before computers, mathematical tables listing all of the primes or prime factorizations up to a given limit were commonly printed.

    Because this decreases exponentially with the number of tests, it provides high confidence although not certainty that a number that passes the repeated test is prime.

    On the other hand, if the test ever fails, then the number is certainly composite. In contrast, some other algorithms guarantee that their answer will always be correct: For instance, this is true of trial division.

    The algorithms with guaranteed-correct output include both deterministic non-random algorithms, such as the AKS primality test , [] and randomized Las Vegas algorithms where the random choices made by the algorithm do not affect its final answer, such as some variations of elliptic curve primality proving.

    The AKS primality test has mathematically proven time complexity, but is slower than elliptic curve primality proving in practice.

    The following table lists some of these tests. In addition to the aforementioned tests that apply to any natural number, some numbers of a special form can be tested for primality more quickly.

    For example, the Lucas—Lehmer primality test can determine whether a Mersenne number one less than a power of two is prime, deterministically, in the same time as a single iteration of the Miller—Rabin test.

    The following table gives the largest known primes of various types. Some of these primes have been found using distributed computing. It is significantly more difficult than primality testing, [] and although many factorization algorithms are known, they are slower than the fastest primality testing methods.

    As with primality testing, there are also factorization algorithms that require their input to have a special form, including the special number field sieve.

    Shor's algorithm can factor any integer in a polynomial number of steps on a quantum computer. Several public-key cryptography algorithms, such as RSA and the Diffie—Hellman key exchange , are based on large prime numbers bit primes are common.

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