# Prime number

primeprime factorprime numbersprimesprimalityprime factorsprime divisorodd primefactorizationfactors
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.wikipedia
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### Number theory

number theoristcombinatorial number theorytheory of numbers
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.
Number theorists study prime numbers as well as the properties of objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers).

### Fundamental theorem of arithmetic

Canonical representation of a positive integerunique factorizationunique factorization theorem
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.
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.

### Miller–Rabin primality test

Miller-Rabin primality testMiller–RabinMiller–Rabin test
Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.
The Miller–Rabin primality test or Rabin–Miller primality test is a primality test: an algorithm which determines whether a given number is prime, similar to the Fermat primality test and the Solovay–Strassen primality test.

### Prime number theorem

distribution of primesdistribution of prime numbersprime number theorem for arithmetic progressions
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 number being prime is inversely proportional to its number of digits, that is, to its logarithm.
In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers.

### Euclid's theorem

infinitude of primesinfinitude of prime numbersinfinitude of the prime numbers
There are infinitely many primes, as demonstrated by Euclid around 300 BC.
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers.

### Factorization

factoringfactorfactors
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.
They proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be further factored into integers greater than 1.

### 7

seven7 (number)
:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

### Natural number

natural numberspositive integerpositive integers
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory.

### Integer factorization

prime factorizationfactoringfactorization
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.
If these factors are further restricted to prime numbers, the process is called prime factorization.

### 5

fivenumber 5
:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Five is the third prime number.

### Analytic number theory

analyticanalytic number theoristanalytic techniques
Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers.
It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem).

### Composite number

compositecomposite numberscomposite integer
A natural number greater than 1 that is not prime is called a composite number.
Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit.

### 17 (number)

17XVIIseventeen
:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
It is a prime number.

### AKS primality test

AKS algorithmComposite numbercyclotomic AKS test
Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.
The algorithm was the first to determine whether any given number is prime or composite within polynomial time.

### Goldbach's conjecture

Goldbach conjectureGoldbach's strong conjectureGoldbach’s conjecture
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 having just one even number between them.
He considered 1 to be a prime number, a convention subsequently abandoned.

### Logarithm

logarithmsloglogarithmic function
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 number being prime is inversely proportional to its number of digits, that is, to its logarithm.
They help describing frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting.

### 2

bracetwo2 (number)
:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Two is the smallest prime number, and the only even prime number (for this reason it is sometimes called "the oddest prime").

### 23 (number)

23twenty-threenumber 23
:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

### 19 (number)

19nineteen19th
:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
It is a prime number.

### 13 (number)

13thirteennumber 13
:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

### 11 (number)

11elevenXI
:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
11 is a prime number.

### 29 (number)

29twenty-nineXXIX
:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

### 41 (number)

41
:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

### 47 (number)

47the number 4747 as an in-joke
:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
It is a prime number, and appears in popular culture as the adopted favorite number of Pomona College and an obsession of the hip hop collective Pro Era.

### 59 (number)

59
:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Fifty-nine is the 17th prime number.