In mathematics, a prime number (or a prime ) is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:
An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any nonzero natural number n can be factored into primes, written as a product of primes or powers of 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 n can be done by trial division, that is to say dividing n by all smaller numbers m , thereby checking whether n is a multiple of m , and therefore not prime but a composite. For big primes, increasingly sophisticated algorithms which are faster than that technique have been devised.
There is no known formula yielding all primes 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 n is prime is inversely proportional to its number of digits, or the logarithm of n . This statement has been proved at 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, many fundamental questions around prime numbers remain open. For example, Goldbach's conjecture which asserts that any even natural number bigger than two is the sum of two primes, or the twin prime conjecture which says that there are infinitely many twin primes (pairs of primes whose difference is two), 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. As of 2009, the largest known prime has about 13 million decimal digits.
Prime numbers and the fundamental theorem of arithmetic
Main article: Fundamental theorem of arithmeticA natural number is called a prime , a prime number or just prime if it has exactly two distinct divisors. Otherwise it is called composite. Therefore, 1 is not prime, since it has only one divisor, namely 1. However, 2 and 3 are prime, since they have exactly two divisors, namely 1 and 2, and 1 and 3, respectively. Next, 4, is composite, since it has 3 divisors: 1, 2, and 4.
Using symbols, a number n > 1 is prime if it cannot be written as a product of two factors a and b , both of which are larger than 1:
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, we can write
As in this example, the same prime factor may occur multiple times. A decomposition
of a number n into (finitely many) prime factors p 1 , p 2 , ... to p t is called prime factorization of n . 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.
The set of all primes is often denoted P .
Examples and first properties
The only even prime number is 2, since any larger even number is divisible by 2. Therefore, the term odd prime refers to any prime number greater than 2.
The image at the right shows a graphical way to show that 12 is not prime. More generally, all prime numbers except 2 and 5, written in the usual decimal system, end in 1, 3, 7 or 9, since numbers ending in 0, 2, 4, 6 or 8 are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. Similarly, all prime numbers above 3 are of the form 6 n − 1 or 6 n + 1, because all other numbers are divisible by 2 or 3. Generalizing this, all prime numbers above q are of form q #· n + m , where 0 < m < q , and m has no prime factor ≤ q .
If p is a prime number and p divides a product ab of integers, then p divides a or p divides b . This proposition is known as Euclid's lemma. It is used in some proofs of the uniqueness of prime factorizations.
Primality of one
The importance of this theorem is one of the reasons for the exclusion of 1 from the set of prime numbers. If 1 were admitted as a prime, the precise statement of the theorem would require additional qualifications, since 3 could then be decomposed in different ways
Until the 19th century, most mathematicians considered the number 1 a prime, the definition being just that a prime is divisible only by 1 and itself but not requiring a specific number of distinct divisors. There is still a large body of mathematical work that is valid despite labeling 1 a prime, such as the work of Stern and Zeisel. 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. The change in label occurred so that the fundamental theorem of arithmetic, as stated, is valid, i.e. , “each number has a unique factorization into primes.” 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.
History
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 2 2 n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 2 16 + 1). However, the very next Fermat number 2 32 + 1 is composite (one of its prime factors is 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 − 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 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + … is divergent. In 1747 he showed that the even perfect numbers are precisely the integers of the form 2 p −1 (2 p − 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 t
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