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A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 5 or 5 1, involve 5 itself. However, 4 is composite because it is a product (2 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen large number being prime is inversely proportional to its number of digits, that is, to its logarithm.
Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes that differ by two. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals.
A natural number (1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it is greater than 1 and cannot be written as the product of two smaller natural numbers. The numbers greater than 1 that are not prime are called composite numbers.[2] In other words, n \displaystyle n is prime if n \displaystyle n items cannot be divided up into smaller equal-size groups of more than one item,[3] or if it is not possible to arrange n \displaystyle n dots into a rectangular grid that is more than one dot wide and more than one dot high.[4] For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers,[5] as there are no other numbers that divide them evenly (without a remainder). 1 is not prime, as it is specifically excluded in the definition. 4 = 2 2 and 6 = 2 3 are both composite.
No even number n \displaystyle n greater than 2 is prime because any such number can be expressed as the product 2 n / 2 \displaystyle 2\times n/2 . Therefore, every prime number other than 2 is an odd number, and is called an odd prime.[9] Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5.[10]
Since 1951 all the largest known primes have been found using these tests on computers.[a] The search for ever larger primes has generated interest outside mathematical circles, through the Great Internet Mersenne Prime Search and other distributed computing projects.[8][29] The idea that prime numbers had few applications outside of pure mathematics[b] was shattered in the 1970s when public-key cryptography and the RSA cryptosystem were invented, using prime numbers as their basis.[32]
Most early Greeks did not even consider 1 to be a number,[36][37] so they could not consider its primality. A few scholars in the Greek and later Roman tradition, including Nicomachus, Iamblichus, Boethius, and Cassiodorus also considered the prime numbers to be a subdivision of the odd numbers, so they did not consider 2 to be prime either. 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.[36] By the Middle Ages and Renaissance, mathematicians began treating 1 as a number, and some of them included it as the first prime number.[38] In the mid-18th century Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be prime.[39] In the 19th century many mathematicians still considered 1 to be prime,[40] and lists of primes that included 1 continued to be published as recently as 1956.[41][42]
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 any number of copies of 1.[40] Similarly, the sieve of Eratosthenes would not work correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1.[42] Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime numbers than they are for 1.[43] By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "unit".[40]
The terms in the product are called prime factors. The same prime factor may occur more than once; this example has two copies of the prime factor 3. \displaystyle 3. When a prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number: for example, in the second way of writing the product above, 3 2 \displaystyle 3^2 denotes the square or second power of 3. \displaystyle 3.
The central importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic.[44] This theorem states that every integer larger than 1 can be written as a product of one or more primes. 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.[45] So, although there are many different ways of finding a factorization using an integer factorization algorithm, they all must produce the same result. Primes can thus be considered the "basic building blocks" of the natural numbers.[46]
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,[49] Furstenberg's proof using general topology,[50] and Kummer's elegant proof.[51]
with one or more prime factors. N \displaystyle N is evenly divisible by each of these factors, but N \displaystyle N has a remainder of one when divided by any of the prime numbers in the given list, so none of the prime factors of N \displaystyle N can be in the given list. Because there is no finite list of all the primes, there must be infinitely many primes.
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.[54] However, there are numerous expressions that do encode all primes, or only primes. One possible formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.[55] There is also a set of Diophantine equations in nine 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.[54]
Other examples of prime-generating formulas come from Mills' theorem and a theorem of Wright. These assert that there are real constants A > 1 \displaystyle A>1 and μ \displaystyle \mu such that
The distribution of primes in the large, such as the question how many primes are smaller than a given, large threshold, is described by the prime number theorem, but no efficient formula for the n \displaystyle n -th prime is known.Dirichlet's theorem on arithmetic progressions, in its basic form, asserts that linear polynomials
with relatively prime integers a \displaystyle a and b \displaystyle b take infinitely many prime values. Stronger forms of the theorem state that the sum of the reciprocals of these prime values diverges, and that different linear polynomials with the same b \displaystyle b have approximately the same proportions of primes.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.
does not grow to infinity as n \displaystyle n goes to infinity (see the Basel problem). In this sense, prime numbers occur more often than squares of natural numbers,although both sets are infinite.[75] Brun's theorem states that the sum of the reciprocals of twin primes,
is an infinite arithmetic progression with modulus 9. 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
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