Prime number

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Composite numbers can be arranged into rectangles but prime numbers cannot.

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.

The property of being prime is called primality. A simple but slow method of checking the primality of a given number ${\displaystyle n}$, called trial division, tests whether ${\displaystyle n}$ is a multiple of any integer between 2 and ${\displaystyle \sqrt{n}}$. 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. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number is a Mersenne prime with 24,862,048 decimal digits.

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.

Definition and examples

Main article: List of prime numbers

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. In other words, ${\displaystyle n}$ is prime if ${\displaystyle n}$ items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange ${\displaystyle n}$ dots into a rectangular grid that is more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers, 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.

Demonstration, with Cuisenaire rods, that 7 is prime, because none of 2, 3, 4, 5, or 6 divide it evenly

The divisors of a natural number ${\displaystyle n}$ are the natural numbers that divide ${\displaystyle n}$ evenly. Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This leads to an equivalent definition of prime numbers: they are the numbers with exactly two positive divisors. Those two are 1 and the number itself. As 1 has only one divisor, itself, it is not prime by this definition. Yet another way to express the same thing is that a number ${\displaystyle n}$ is prime if it is greater than one and if none of the numbers ${\displaystyle 2,3,\dots ,n-1}$ divides ${\displaystyle n}$ evenly.

The first 25 prime numbers (all the prime numbers less than 100) are:

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 (sequence A000040 in the OEIS).

No even number ${\displaystyle n}$ greater than 2 is prime because any such number can be expressed as the product ${\displaystyle 2\times n/2}$. Therefore, every prime number other than 2 is an odd number, and is called an odd prime. 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.

The set of all primes is sometimes denoted by ${\displaystyle \mathbf{P}}$ (a boldface capital P) or by ${\displaystyle \mathbb{P}}$ (a blackboard bold capital P).

History

The Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from ancient Greek mathematics. Euclid's Elements (c. 300 BC) proves the infinitude of primes and the fundamental theorem of arithmetic, and shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes.

Around 1000 AD, the Islamic mathematician Ibn al-Haytham (Alhazen) found Wilson's theorem, characterizing the prime numbers as the numbers ${\displaystyle n}$ that evenly divide ${\displaystyle (n-1)!+1}$. He also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it. Another Islamic mathematician, Ibn al-Banna' al-Marrakushi, observed that the sieve of Eratosthenes can be sped up by considering only the prime divisors up to the square root of the upper limit. Fibonacci took the innovations from Islamic mathematics to Europe. His book Liber Abaci (1202) was the first to describe trial division for testing primality, again using divisors only up to the square root.

In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also investigated the primality of the Fermat numbers ${\displaystyle 2^{2^n}+1}$, and Marin Mersenne studied the Mersenne primes, prime numbers of the form ${\displaystyle 2^p-1}$ with ${\displaystyle p}$ itself a prime. Christian Goldbach formulated Goldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler. Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be constructed from Mersenne primes. He introduced methods from mathematical analysis to this area in his proofs of the infinitude of the primes and the divergence of the sum of the reciprocals of the primes ${\displaystyle \frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\cdots }$. At the start of the 19th century, Legendre and Gauss conjectured that as ${\displaystyle x}$ tends to infinity, the number of primes up to ${\displaystyle x}$ is asymptotic to ${\displaystyle x/\log x}$, where ${\displaystyle \log x}$ is the natural logarithm of ${\displaystyle x}$. A weaker consequence of this high density of primes was Bertrand's postulate, that for every ${\displaystyle n>1}$ there is a prime between ${\displaystyle n}$ and ${\displaystyle 2n}$, proved in 1852 by Pafnuty Chebyshev. Ideas of Bernhard Riemann in his 1859 paper on the zeta-function sketched an outline for proving the conjecture of Legendre and Gauss. Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by Hadamard and de la Vallée Poussin, and the result is now known as the prime number theorem. Another important 19th century result was Dirichlet's theorem on arithmetic progressions, that certain arithmetic progressions contain infinitely many primes.

Many mathematicians have worked on primality tests for numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms include Pépin's test for Fermat numbers (1877), Proth's theorem (c. 1878), the Lucas–Lehmer primality test (originated 1856), and the generalized Lucas primality test.

Since 1951 all the largest known primes have been found using these tests on computers. The search for ever larger primes has generated interest outside mathematical circles, through the Great Internet Mersenne Prime Search and other distributed computing projects. The idea that prime numbers had few applications outside of pure mathematics was shattered in the 1970s when public-key cryptography and the RSA cryptosystem were invented, using prime numbers as their basis.

The increased practical importance of computerized primality testing and factorization led to the development of improved methods capable of handling large numbers of unrestricted form. The mathematical theory of prime numbers also moved forward with the Green–Tao theorem (2004) that there are arbitrarily long arithmetic progressions of prime numbers, and Yitang Zhang's 2013 proof that there exist infinitely many prime gaps of bounded size.

Primality of one

Most early Greeks did not even consider 1 to be a number, 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. By the Middle Ages and Renaissance, mathematicians began treating 1 as a number, and some of them included it as the first prime number. 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. In the 19th century many mathematicians still considered 1 to be prime, and lists of primes that included 1 continued to be published as recently as 1956.

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. 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. 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. 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".

Elementary properties

Unique factorization

Main article: Fundamental theorem of arithmetic

Writing a number as a product of prime numbers is called a prime factorization of the number. For example:

${\displaystyle \begin{array}{rl}34866& =2\times 3\times 3\times 13\times 149\\ & =2\times 3^2\times 13\times 149.\end{array}}$

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 ${\displaystyle 3.}$ 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, ${\displaystyle 3^2}$ denotes the square or second power of ${\displaystyle 3.}$

The central importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic. 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. 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.

Some proofs of the uniqueness of prime factorizations are based on Euclid's lemma: If ${\displaystyle p}$ is a prime number and ${\displaystyle p}$ divides a product ${\displaystyle ab}$ of integers ${\displaystyle a}$ and ${\displaystyle b,}$ then ${\displaystyle p}$ divides ${\displaystyle a}$ or ${\displaystyle p}$ divides ${\displaystyle b}$ (or both). Conversely, if a number ${\displaystyle p}$ has the property that when it divides a product it always divides at least one factor of the product, then ${\displaystyle p}$ must be prime.

Infinitude

Main article: Euclid's theorem

There are infinitely many prime numbers. Another way of saying this is that the sequence

2, 3, 5, 7, 11, 13, ...

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, Furstenberg's proof using general topology, and Kummer's elegant proof.

Euclid's proof shows that every finite list of primes is incomplete. The key idea is to multiply together the primes in any given list and add ${\displaystyle 1.}$ If the list consists of the primes ${\displaystyle p_1,p_2,\dots ,p_n,}$ this gives the number

${\displaystyle N=1+p_1\cdot p_2\cdots p_n.}$

By the fundamental theorem, ${\displaystyle N}$ has a prime factorization

${\displaystyle N=p_1^{\prime }\cdot p_2^{\prime }\cdots p_m^{\prime }}$

with one or more prime factors. ${\displaystyle N}$ is evenly divisible by each of these factors, but ${\displaystyle N}$ has a remainder of one when divided by any of the prime numbers in the given list, so none of the prime factors of ${\displaystyle N}$ can be in the given list. 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. The first five of them are prime, but the sixth,

${\displaystyle 1+(2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13)=30031=59\cdot 509,}$

is a composite number.

Formulas for primes

Main article: Formula for 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. 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. 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.

Other examples of prime-generating formulas come from Mills' theorem and a theorem of Wright. These assert that there are real constants ${\displaystyle A>1}$ and ${\displaystyle \mu }$ such that

${\displaystyle \lfloor A^{3^n}\rfloor \text{ and }\lfloor 2^{{\cdots }^{2^{2^{\mu }}}}\rfloor }$

are prime for any natural number ${\displaystyle n}$ in the first formula, and any number of exponents in the second formula. Here ${\displaystyle \lfloor \cdot \rfloor }$ represents the floor function, the largest integer less than or equal to the number in question. However, these are not useful for generating primes, as the primes must be generated first in order to compute the values of ${\displaystyle A}$ or ${\displaystyle \mu .}$

Open questions

Further information: Category:Conjectures about prime numbers

Many conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer ${\displaystyle n}$ greater than 2 can be written as a sum of two primes. As of 2014, this conjecture has been verified for all numbers up to ${\displaystyle n=4\cdot {10}^{18}.}$ Weaker statements than this have been proven, for example, Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime (the product of two primes). Also, any even integer greater than 10 can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.

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 ${\displaystyle n!+2,n!+3,\dots ,n!+n}$ consists of ${\displaystyle n-1}$ composite numbers, for any natural number ${\displaystyle n.}$ However, large prime gaps occur much earlier than this argument shows. For example, the first prime gap of length 8 is between the primes 89 and 97, much smaller than ${\displaystyle 8!=40320.}$ It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2; this is the twin prime conjecture. Polignac's conjecture states more generally that for every positive integer ${\displaystyle k,}$ there are infinitely many pairs of consecutive primes that differ by ${\displaystyle 2k.}$ Andrica's conjecture, Brocard's conjecture, Legendre's conjecture, and Oppermann's conjecture all suggest that the largest gaps between primes from ${\displaystyle 1}$ to ${\displaystyle n}$ should be at most approximately ${\displaystyle \sqrt{n},}$ a result that is known to follow from the Riemann hypothesis, while the much stronger Cramér conjecture sets the largest gap size at ${\displaystyle O((\log n{)}^2).}$ Prime gaps can be generalized to prime ${\displaystyle k}$-tuples, patterns in the differences among more than two prime numbers. 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 properties

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. The problem asked for the value of the infinite sum ${\displaystyle 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\dots ,}$ which today can be recognized as the value ${\displaystyle \zeta (2)}$ of the Riemann zeta function. This function is closely connected to the prime numbers and to one of the most significant unsolved problems in mathematics, the Riemann hypothesis. Euler showed that ${\displaystyle \zeta (2)={\pi }^2/6}$. The reciprocal of this number, ${\displaystyle 6/{\pi }^2}$, is the limiting probability that two random numbers selected uniformly from a large range are relatively prime (have no factors in common).

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 ${\displaystyle n}$-th prime is known. Dirichlet's theorem on arithmetic progressions, in its basic form, asserts that linear polynomials

${\displaystyle p(n)=a+bn}$

with relatively prime integers ${\displaystyle a}$ and ${\displaystyle b}$ 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 ${\displaystyle b}$ 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.

Analytical proof of Euclid's theorem

Euler's proof that there are infinitely many primes considers the sums of reciprocals of primes,

${\displaystyle \frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots +\frac{1}{p}.}$

Euler showed that, for any arbitrary real number ${\displaystyle x}$, there exists a prime ${\displaystyle p}$ for which this sum is bigger than ${\displaystyle x}$. This shows that there are infinitely many primes, because if there were finitely many primes the sum would reach its maximum value at the biggest prime rather than growing past every ${\displaystyle x}$. The growth rate of this sum is described more precisely by Mertens' second theorem. For comparison, the sum

${\displaystyle \frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots +\frac{1}{n^2}}$

does not grow to infinity as ${\displaystyle n}$ 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. Brun's theorem states that the sum of the reciprocals of twin primes,

${\displaystyle \left(\frac{1}{3}+\frac{1}{5}\right)+\left(\frac{1}{5}+\frac{1}{7}\right)+\left(\frac{1}{11}+\frac{1}{13}\right)+\cdots ,}$

is finite. 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.

Number of primes below a given bound

Main articles: Prime number theorem and Prime-counting function

The relative error of ${\displaystyle \frac{n}{\log n}}$ and the logarithmic integral ${\displaystyle \mathrm{Li}(n)}$ as approximations to the prime-counting function. Both relative errors decrease to zero as ${\displaystyle n}$ grows, but the convergence to zero is much more rapid for the logarithmic integral.

The prime-counting function ${\displaystyle \pi (n)}$ is defined as the number of primes not greater than ${\displaystyle n}$. For example, ${\displaystyle \pi (11)=5}$, since there are five primes less than or equal to 11. Methods such as the Meissel–Lehmer algorithm can compute exact values of ${\displaystyle \pi (n)}$ faster than it would be possible to list each prime up to ${\displaystyle n}$. The prime number theorem states that ${\displaystyle \pi (n)}$ is asymptotic to ${\displaystyle n/\log n}$, which is denoted as

${\displaystyle \pi (n)\sim \frac{n}{\log n},}$

and means that the ratio of ${\displaystyle \pi (n)}$ to the right-hand fraction approaches 1 as ${\displaystyle n}$ grows to infinity. This implies that the likelihood that a randomly chosen number less than ${\displaystyle n}$ is prime is (approximately) inversely proportional to the number of digits in ${\displaystyle n}$. It also implies that the ${\displaystyle n}$th prime number is proportional to ${\displaystyle n\log n}$ and therefore that the average size of a prime gap is proportional to ${\displaystyle \log n}$. A more accurate estimate for ${\displaystyle \pi (n)}$ is given by the offset logarithmic integral

${\displaystyle \pi (n)\sim \mathrm{Li}(n)={\int }_2^n\frac{dt}{\log t}.}$

Arithmetic progressions

Main articles: Dirichlet's theorem on arithmetic progressions and Green–Tao theorem

An arithmetic progression is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference. This difference is called the modulus of the progression. For example,

3, 12, 21, 30, 39, ...,

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

${\displaystyle a,a+q,a+2q,a+3q,\dots }$

can have more than one prime only when its remainder ${\displaystyle a}$ and modulus ${\displaystyle q}$ are relatively prime. If they are relatively prime, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes.

Primes in the arithmetic progressions modulo 9. Each row of the thin horizontal band shows one of the nine possible progressions mod 9, with prime numbers marked in red. The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number (the number 3); the remaining progressions of numbers that are 2, 4, 5, 7, and 8 mod 9 have infinitely many prime numbers, with similar numbers of primes in each progression.

The Green–Tao theorem shows that there are arbitrarily long finite arithmetic progressions consisting only of primes.

Prime values of quadratic polynomials

The Ulam spiral. Prime numbers (red) cluster on some diagonals and not others. Prime values of ${\displaystyle 4n^2-2n+41}$ are shown in blue.

Euler noted that the function

${\displaystyle n^2-n+41}$

yields prime numbers for ${\displaystyle 1\le n\le 40}$, although composite numbers appear among its later values. The search for an explanation for this phenomenon led to the deep algebraic number theory of Heegner numbers and the class number problem. The Hardy-Littlewood conjecture F predicts the density of primes among the values of quadratic polynomials with integer coefficients in terms of the logarithmic integral and the polynomial coefficients. 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.

Zeta function and the Riemann hypothesis

Main article: Riemann hypothesis

Plot of the absolute values of the zeta function, showing some of its features

One of the most famous unsolved questions in mathematics, dating from 1859, and one of the Millennium Prize Problems, is the Riemann hypothesis, which asks where the zeros of the Riemann zeta function ${\displaystyle \zeta (s)}$ are located. This function is an analytic function on the complex numbers. For complex numbers ${\displaystyle s}$ with real part greater than one it equals both an infinite sum over all integers, and an infinite product over the prime numbers,

${\displaystyle \zeta (s)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}=\prod _{p\text{ prime}}\frac{1}{1-p^{-s}}.}$

This equality between a sum and a product, discovered by Euler, is called an Euler product. The Euler product can be derived from the fundamental theorem of arithmetic, and shows the close connection between the zeta function and the prime numbers. It leads to another proof that there are infinitely many primes: if there were only finitely many, then the sum-product equality would also be valid at ${\displaystyle s=1}$, but the sum would diverge (it is the harmonic series ${\displaystyle 1+\frac{1}{2}+\frac{1}{3}+\dots }$) while the product would be finite, a contradiction.

The Riemann hypothesis states that the zeros of the zeta-function are all either negative even numbers, or complex numbers with real part equal to 1/2. The original proof of the prime number theorem was based on a weak form of this hypothesis, that there are no zeros with real part equal to 1, although other more elementary proofs have been found. The prime-counting function can be expressed by Riemann's explicit formula as a sum in which each term comes from one of the zeros of the zeta function; the main term of this sum is the logarithmic integral, and the remaining terms cause the sum to fluctuate above and below the main term. In this sense, the zeros control how regularly the prime numbers are distributed. If the Riemann hypothesis is true, these fluctuations will be small, and the asymptotic distribution of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of ${\displaystyle x}$ for intervals near a number ${\displaystyle x}$).

Abstract algebra

Modular arithmetic and finite fields

Main article: Modular arithmetic

Modular arithmetic modifies usual arithmetic by only using the numbers ${\displaystyle \{0,1,2,\dots ,n-1\}}$, for a natural number ${\displaystyle n}$ called the modulus. Any other natural number can be mapped into this system by replacing it by its remainder after division by ${\displaystyle n}$. Modular sums, differences and products are calculated by performing the same replacement by the remainder on the result of the usual sum, difference, or product of integers. Equality of integers corresponds to congruence in modular arithmetic: ${\displaystyle x}$ and ${\displaystyle y}$ are congruent (written ${\displaystyle x\equiv y}$ mod ${\displaystyle n}$) when they have the same remainder after division by ${\displaystyle n}$. However, in this system of numbers, division by all nonzero numbers is possible if and only if the modulus is prime. For instance, with the prime number ${\displaystyle 7}$ as modulus, division by ${\displaystyle 3}$ is possible: ${\displaystyle 2/3\equiv 3mod7}$, because clearing denominators by multiplying both sides by ${\displaystyle 3}$ gives the valid formula ${\displaystyle 2\equiv 9mod7}$. However, with the composite modulus ${\displaystyle 6}$, division by ${\displaystyle 3}$ is impossible. There is no valid solution to ${\displaystyle 2/3\equiv xmod6}$: clearing denominators by multiplying by ${\displaystyle 3}$ causes the left-hand side to become ${\displaystyle 2}$ while the right-hand side becomes either ${\displaystyle 0}$ or ${\displaystyle 3}$. 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. For instance, Fermat's little theorem states that if ${\displaystyle a\not\equiv 0}$ (mod ${\displaystyle p}$), then ${\displaystyle a^{p-1}\equiv 1}$ (mod ${\displaystyle p}$). Summing this over all choices of ${\displaystyle a}$ gives the equation

${\displaystyle \sum _{a=1}^{p-1}{a}^{p-1}\equiv (p-1)\cdot 1\equiv -1(\mathrm{mod}p),}$

valid whenever ${\displaystyle p}$ is prime. Giuga's conjecture says that this equation is also a sufficient condition for ${\displaystyle p}$ to be prime. Wilson's theorem says that an integer ${\displaystyle p>1}$ is prime if and only if the factorial ${\displaystyle (p-1)!}$ is congruent to ${\displaystyle -1}$ mod ${\displaystyle p}$. For a composite number ${\displaystyle n=r\cdot s}$ this cannot hold, since one of its factors divides both n and ${\displaystyle (n-1)!}$, and so ${\displaystyle (n-1)!\equiv -1(\mathrm{mod}n)}$ is impossible.

p-adic numbers

Main article: p-adic number

The ${\displaystyle p}$-adic order ${\displaystyle {\nu }_p(n)}$ of an integer ${\displaystyle n}$ is the number of copies of ${\displaystyle p}$ in the prime factorization of ${\displaystyle n}$. The same concept can be extended from integers to rational numbers by defining the ${\displaystyle p}$-adic order of a fraction ${\displaystyle m/n}$ to be ${\displaystyle {\nu }_p(m)-{\nu }_p(n)}$. The ${\displaystyle p}$-adic absolute value ${\displaystyle |q{|}_p}$ of any rational number ${\displaystyle q}$ is then defined as ${\displaystyle |q{|}_p=p^{-{\nu }_p(q)}}$. Multiplying an integer by its ${\displaystyle p}$-adic absolute value cancels out the factors of ${\displaystyle p}$ in its factorization, leaving only the other primes. Just as the distance between two real numbers can be measured by the absolute value of their distance, the distance between two rational numbers can be measured by their ${\displaystyle p}$-adic distance, the ${\displaystyle p}$-adic absolute value of their difference. For this definition of distance, two numbers are close together (they have a small distance) when their difference is divisible by a high power of ${\displaystyle p}$. In the same way that the real numbers can be formed from the rational numbers and their distances, by adding extra limiting values to form a complete field, the rational numbers with the ${\displaystyle p}$-adic distance can be extended to a different complete field, the ${\displaystyle p}$-adic numbers.

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 extension from the rational numbers to the real numbers, for instance, is a place in which the distance between numbers is the usual absolute value of their difference. 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. According to Ostrowski's theorem, up to a natural notion of equivalence, the real numbers and ${\displaystyle p}$-adic numbers, with their orders and absolute values, are the only valuations, absolute values, and places on the rational numbers. The local-global principle allows certain problems over the rational numbers to be solved by piecing together solutions from each of their places, again underlining the importance of primes to number theory.

Prime elements in rings

Main articles: Prime element and Irreducible element

The Gaussian primes with norm less than 500

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 ${\displaystyle p}$ of a ring ${\displaystyle R}$ is called prime if it is nonzero, has no multiplicative inverse (that is, it is not a unit), and satisfies the following requirement: whenever ${\displaystyle p}$ divides the product ${\displaystyle xy}$ of two elements of ${\displaystyle R}$, it also divides at least one of ${\displaystyle x}$ or ${\displaystyle y}$. 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,

${\displaystyle \{\dots ,-11,-7,-5,-3,-2,2,3,5,7,11,\dots \}.}$

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. An example of such a domain is the Gaussian integers ${\displaystyle \mathbb{Z}[i]}$, the ring of complex numbers of the form ${\displaystyle a+bi}$ where ${\displaystyle i}$ denotes the imaginary unit and ${\displaystyle a}$ and ${\displaystyle b}$ are arbitrary integers. Its prime elements are known as Gaussian primes. Not every number that is prime among the integers remains prime in the Gaussian integers; for instance, the number 2 can be written as a product of the two Gaussian primes ${\displaystyle 1+i}$ and ${\displaystyle 1-i}$. 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. This is a consequence of Fermat's theorem on sums of two squares, which states that an odd prime ${\displaystyle p}$ is expressible as the sum of two squares, ${\displaystyle p=x^2+y^2}$, and therefore factorable as ${\displaystyle p=(x+iy)(x-iy)}$, exactly when ${\displaystyle p}$ is 1 mod 4.

Prime ideals

Main article: Prime ideals

Not every ring is a unique factorization domain. For instance, in the ring of numbers ${\displaystyle a+b\sqrt{-5}}$ (for integers ${\displaystyle a}$ and ${\displaystyle b}$) the number ${\displaystyle 21}$ has two factorizations ${\displaystyle 21=3\cdot 7=(1+2\sqrt{-5})(1-2\sqrt{-5})}$, where neither of the four factors can be reduced any further, so it does not have a unique factorization. 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. Arithmetic geometry also benefits from this notion, and many concepts exist in both geometry and number theory. 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. Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected with the failure of unique factorization in the cyclotomic integers. The question of how many integer prime numbers factor into a product of multiple prime ideals in an algebraic number field is addressed by Chebotarev's density theorem, which (when applied to the cyclotomic integers) has Dirichlet's theorem on primes in arithmetic progressions as a special case.

Group theory

In the theory of finite groups the Sylow theorems imply that, if a power of a prime number ${\displaystyle p^n}$ divides the order of a group, then the group has a subgroup of order ${\displaystyle p^n}$. By Lagrange's theorem, any group of prime order is a cyclic group, and by Burnside's theorem any group whose order is divisible by only two primes is solvable.

Computational methods

The small gear in this piece of farm equipment has 13 teeth, a prime number, and the middle gear has 21, relatively prime to 13.

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 other than the use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance.

This vision of the purity of number theory was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public-key cryptography algorithms. These applications have led to significant study of algorithms for computing with prime numbers, and in particular of primality testing, methods for determining whether a given number is prime. 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.

Trial division

Main article: Trial division

The most basic method of checking the primality of a given integer ${\displaystyle n}$ is called trial division. This method divides ${\displaystyle n}$ by each integer from 2 up to the square root of ${\displaystyle n}$. Any such integer dividing ${\displaystyle n}$ evenly establishes ${\displaystyle n}$ as composite; otherwise it is prime. Integers larger than the square root do not need to be checked because, whenever ${\displaystyle n=a\cdot b}$, one of the two factors ${\displaystyle a}$ and ${\displaystyle b}$ is less than or equal to the square root of ${\displaystyle n}$. Another optimization is to check only primes as factors in this range. For instance, to check whether 37 is prime, this method divides it by the primes in the range from 2 to ${\displaystyle \sqrt{37}}$, which are 2, 3, and 5. 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. However, trial division is still used, with a smaller limit than the square root on the divisor size, to quickly discover composite numbers with small factors, before using more complicated methods on the numbers that pass this filter.

Sieves

The sieve of Eratosthenes starts with all numbers unmarked (gray). It repeatedly finds the first unmarked number, marks it as prime (dark colors) and marks its square and all later multiples as composite (lighter colors). After marking the multiples of 2 (red), 3 (green), 5 (blue), and 7 (yellow), all primes up to the square root of the table size have been processed, and all remaining unmarked numbers (11, 13, etc.) are marked as primes (magenta).

Main article: Sieve of Eratosthenes

Before computers, mathematical tables listing all of the primes or prime factorizations up to a given limit were commonly printed. The oldest method for generating a list of primes is called the sieve of Eratosthenes. The animation shows an optimized variant of this method. Another more asymptotically efficient sieving method for the same problem is the sieve of Atkin. In advanced mathematics, sieve theory applies similar methods to other problems.

Primality testing versus primality proving

Some of the fastest modern tests for whether an arbitrary given number ${\displaystyle n}$ is prime are probabilistic (or Monte Carlo) algorithms, meaning that they have a small random chance of producing an incorrect answer. For instance the Solovay–Strassen primality test on a given number ${\displaystyle p}$ chooses a number ${\displaystyle a}$ randomly from ${\displaystyle 2}$ through ${\displaystyle p-2}$ and uses modular exponentiation to check whether ${\displaystyle a^{(p-1)/2}\pm 1}$ is divisible by ${\displaystyle p}$. If so, it answers yes and otherwise it answers no. If ${\displaystyle p}$ really is prime, it will always answer yes, but if ${\displaystyle p}$ is composite then it answers yes with probability at most 1/2 and no with probability at least 1/2. If this test is repeated ${\displaystyle n}$ times on the same number, the probability that a composite number could pass the test every time is at most ${\displaystyle 1/2^n}$. 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. A composite number that passes such a test is called a pseudoprime.

In contrast, some other algorithms guarantee that their answer will always be correct: primes will always be determined to be prime and composites will always be determined to be composite. 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. When the elliptic curve method concludes that a number is prime, it provides primality certificate that can be verified quickly. The elliptic curve primality test is the fastest in practice of the guaranteed-correct primality tests, but its runtime analysis is based on heuristic arguments rather than rigorous proofs. The AKS primality test has mathematically proven time complexity, but is slower than elliptic curve primality proving in practice. These methods can be used to generate large random prime numbers, by generating and testing random numbers until finding one that is prime; when doing this, a faster probabilistic test can quickly eliminate most composite numbers before a guaranteed-correct algorithm is used to verify that the remaining numbers are prime.

The following table lists some of these tests. Their running time is given in terms of ${\displaystyle n}$, the number to be tested and, for probabilistic algorithms, the number ${\displaystyle k}$ of tests performed. Moreover, ${\displaystyle \epsilon }$ is an arbitrarily small positive number, and log is the logarithm to an unspecified base. The big O notation means that each time bound should be multiplied by a constant factor to convert it from dimensionless units to units of time; this factor depends on implementation details such as the type of computer used to run the algorithm, but not on the input parameters ${\displaystyle n}$ and ${\displaystyle k}$.

Test	Developed in	Type	Running time	Notes
AKS primality test	2002	deterministic	${\displaystyle O((\log n{)}^{6+\epsilon })}$
Elliptic curve primality proving	1986	Las Vegas	${\displaystyle O((\log n{)}^{4+\epsilon })}$ heuristically
Baillie–PSW primality test	1980	Monte Carlo	${\displaystyle O((\log n{)}^{2+\epsilon })}$
Miller–Rabin primality test	1980	Monte Carlo	${\displaystyle O(k(\log n{)}^{2+\epsilon })}$	error probability ${\displaystyle 4^{-k}}$
Solovay–Strassen primality test	1977	Monte Carlo	${\displaystyle O(k(\log n{)}^{2+\epsilon })}$	error probability ${\displaystyle 2^{-k}}$

Special-purpose algorithms and the largest known prime

Further information: List of prime numbers

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. This is why since 1992 (as of December 2018) the largest known prime has always been a Mersenne prime. It is conjectured that there are infinitely many Mersenne primes.

The following table gives the largest known primes of various types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively.

Type	Prime	Number of decimal digits	Date	Found by
Mersenne prime	282,589,933 − 1	24,862,048	December 7, 2018	Patrick Laroche, Great Internet Mersenne Prime Search
Proth prime	10,223 × 231,172,165 + 1	9,383,761	October 31, 2016	Péter Szabolcs, PrimeGrid
factorial prime	208,003! − 1	1,015,843	July 2016	Sou Fukui
primorial prime[e]	1,098,133# − 1	476,311	March 2012	James P. Burt, PrimeGrid
twin primes	2,996,863,034,895 × 21,290,000 ± 1	388,342	September 2016	Tom Greer, PrimeGrid

Integer factorization

Main article: Integer factorization

Given a composite integer ${\displaystyle n}$, the task of providing one (or all) prime factors is referred to as factorization of ${\displaystyle n}$. It is significantly more difficult than primality testing, and although many factorization algorithms are known, they are slower than the fastest primality testing methods. Trial division and Pollard's rho algorithm can be used to find very small factors of ${\displaystyle n}$, and elliptic curve factorization can be effective when ${\displaystyle n}$ has factors of moderate size. Methods suitable for arbitrary large numbers that do not depend on the size of its factors include the quadratic sieve and general number field sieve. As with primality testing, there are also factorization algorithms that require their input to have a special form, including the special number field sieve. As of December 2019 the largest number known to have been factored by a general-purpose algorithm is RSA-240, which has 240 decimal digits (795 bits) and is the product of two large primes.

Shor's algorithm can factor any integer in a polynomial number of steps on a quantum computer. However, current technology can only run this algorithm for very small numbers. As of October 2012 the largest number that has been factored by a quantum computer running Shor's algorithm is 21.

Other computational applications

Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (2048-bit primes are common). RSA relies on the assumption that it is much easier (that is, more efficient) to perform the multiplication of two (large) numbers ${\displaystyle x}$ and ${\displaystyle y}$ than to calculate ${\displaystyle x}$ and ${\displaystyle y}$ (assumed coprime) if only the product ${\displaystyle xy}$ is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation (computing ${\displaystyle a^{b}modc}$), while the reverse operation (the discrete logarithm) is thought to be a hard problem.

Prime numbers are frequently used for hash tables. For instance the original method of Carter and Wegman for universal hashing was based on computing hash functions by choosing random linear functions modulo large prime numbers. Carter and Wegman generalized this method to ${\displaystyle k}$-independent hashing by using higher-degree polynomials, again modulo large primes. As well as in the hash function, prime numbers are used for the hash table size in quadratic probing based hash tables to ensure that the probe sequence covers the whole table.

Some checksum methods are based on the mathematics of prime numbers. For instance the checksums used in International Standard Book Numbers are defined by taking the rest of the number modulo 11, a prime number. Because 11 is prime this method can detect both single-digit errors and transpositions of adjacent digits. Another checksum method, Adler-32, uses arithmetic modulo 65521, the largest prime number less than ${\displaystyle 2^{16}}$. Prime numbers are also used in pseudorandom number generators including linear congruential generators and the Mersenne Twister.

Other applications

Prime numbers are of central importance to number theory but also have many applications to other areas within mathematics, including abstract algebra and elementary geometry. For example, it is possible to place prime numbers of points in a two-dimensional grid so that no three are in a line, or so that every triangle formed by three of the points has large area. Another example is Eisenstein's criterion, a test for whether a polynomial is irreducible based on divisibility of its coefficients by a prime number and its square.

The connected sum of two prime knots

The concept of a prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field of a given field is its smallest subfield that contains both 0 and 1. It is either the field of rational numbers or a finite field with a prime number of elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the connected sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. The prime decomposition of 3-manifolds is another example of this type.

Beyond mathematics and computing, prime numbers have potential connections to quantum mechanics, and have been used metaphorically in the arts and literature. They have also been used in evolutionary biology to explain the life cycles of cicadas.

Constructible polygons and polygon partitions

Construction of a regular pentagon using straightedge and compass. This is only possible because 5 is a Fermat prime.

Fermat primes are primes of the form

${\displaystyle F_k=2^{2^k}+1,}$

with ${\displaystyle k}$ a nonnegative integer. They are named after Pierre de Fermat, who conjectured that all such numbers are prime. The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime, but ${\displaystyle F_5}$ is composite and so are all other Fermat numbers that have been verified as of 2017. A regular ${\displaystyle n}$-gon is constructible using straightedge and compass if and only if the odd prime factors of ${\displaystyle n}$ (if any) are distinct Fermat primes. Likewise, a regular ${\displaystyle n}$-gon may be constructed using straightedge, compass, and an angle trisector if and only if the prime factors of ${\displaystyle n}$ are any number of copies of 2 or 3 together with a (possibly empty) set of distinct Pierpont primes, primes of the form ${\displaystyle 2^a{3}^b+1}$.

It is possible to partition any convex polygon into ${\displaystyle n}$ smaller convex polygons of equal area and equal perimeter, when ${\displaystyle n}$ is a power of a prime number, but this is not known for other values of ${\displaystyle n}$.

Quantum mechanics

Beginning with the work of Hugh Montgomery and Freeman Dyson in the 1970s, mathematicians and physicists have speculated that the zeros of the Riemann zeta function are connected to the energy levels of quantum systems. Prime numbers are also significant in quantum information science, thanks to mathematical structures such as mutually unbiased bases and symmetric informationally complete positive-operator-valued measures.

Biology

The evolutionary strategy used by cicadas of the genus Magicicada makes use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. Biologists theorize that these prime-numbered breeding cycle lengths have evolved in order to prevent predators from synchronizing with these cycles. In contrast, the multi-year periods between flowering in bamboo plants are hypothesized to be smooth numbers, having only small prime numbers in their factorizations.

Arts and literature

Prime numbers have influenced many artists and writers. The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". In works such as La Nativité du Seigneur (1935) and Quatre études de rythme (1949–50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in the third étude, "Neumes rythmiques". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations".

In his science fiction novel Contact, scientist Carl Sagan suggested that prime factorization could be used as a means of establishing two-dimensional image planes in communications with aliens, an idea that he had first developed informally with American astronomer Frank Drake in 1975. In the novel The Curious Incident of the Dog in the Night-Time by Mark Haddon, the narrator arranges the sections of the story by consecutive prime numbers as a way to convey the mental state of its main character, a mathematically gifted teen with Asperger syndrome. Prime numbers are used as a metaphor for loneliness and isolation in the Paolo Giordano novel The Solitude of Prime Numbers, in which they are portrayed as "outsiders" among integers.

Aorta

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Aorta

 

Aorta
Schematic view of the aorta and its segments
Branches of the aorta
Details
Pronunciation	/eɪˈɔːrtə/
Precursor	Truncus arteriosus, fourth left branchial artery, paired dorsal aortae (combine into the single descending aorta)
Source	Left ventricle
Branches	Ascending aorta: Right and left coronary arteries Arch of aorta (supra-aortic vessels): Brachiocephalic trunk Left common carotid artery Left subclavian artery Descending aorta, thoracic part: Left bronchial arteries Esophageal arteries to the thoracic part of the esophagus Third to eleventh posterior intercostal arteries and the subcostal arteries Descending aorta, abdominal part: Parietal branches: Inferior phrenic arteries Lumbar arteries Median sacral artery Visceral branches: Celiac trunk Middle suprarenal arteries Superior mesenteric artery Renal arteries Gonadal arteries (testicular in males, ovarian in females) Inferior mesenteric artery Terminal branches: Common iliac arteries Median sacral artery
Vein	Combination of coronary sinus, superior vena cava and inferior vena cava
Supplies	The systemic circulation (entire body with exception of the respiratory zone of the lung which is supplied by the pulmonary circulation)
Identifiers
Latin	Aorta, arteria maxima
MeSH	D001011
TA98	A12.2.02.001
TA2	4175
FMA	3734

The aorta (/eɪˈɔːrtə/ ay-OR-tə; PL: aortas or aortae) is the main and largest artery in the human body, originating from the left ventricle of the heart, branching upwards immediately after, and extending down to the abdomen, where it splits at the aortic bifurcation into two smaller arteries (the common iliac arteries). The aorta distributes oxygenated blood to all parts of the body through the systemic circulation.

Structure

Sections

Course of the aorta in the thorax (anterior view), starting posterior to the main pulmonary artery, then anterior to the right pulmonary arteries, the trachea and the esophagus, then turning posteriorly to course dorsally to these structures.

In anatomical sources, the aorta is usually divided into sections.

One way of classifying a part of the aorta is by anatomical compartment, where the thoracic aorta (or thoracic portion of the aorta) runs from the heart to the diaphragm. The aorta then continues downward as the abdominal aorta (or abdominal portion of the aorta) from the diaphragm to the aortic bifurcation.

Another system divides the aorta with respect to its course and the direction of blood flow. In this system, the aorta starts as the ascending aorta, travels superiorly from the heart, and then makes a hairpin turn known as the aortic arch. Following the aortic arch, the aorta then travels inferiorly as the descending aorta. The descending aorta has two parts. The aorta begins to descend in the thoracic cavity and is consequently known as the thoracic aorta. After the aorta passes through the diaphragm, it is known as the abdominal aorta. The aorta ends by dividing into two major blood vessels, the common iliac arteries and a smaller midline vessel, the median sacral artery.

Ascending aorta

Main article: Ascending aorta

The ascending aorta begins at the opening of the aortic valve in the left ventricle of the heart. It runs through a common pericardial sheath with the pulmonary trunk. These two blood vessels twist around each other, causing the aorta to start out posterior to the pulmonary trunk, but end by twisting to its right and anterior side. The transition from ascending aorta to aortic arch is at the pericardial reflection on the aorta.

At the root of the ascending aorta, the lumen has three small pockets between the cusps of the aortic valve and the wall of the aorta, which are called the aortic sinuses or the sinuses of Valsalva. The left aortic sinus contains the origin of the left coronary artery and the right aortic sinus likewise gives rise to the right coronary artery. Together, these two arteries supply the heart. The posterior aortic sinus does not give rise to a coronary artery. For this reason the left, right and posterior aortic sinuses are also called left-coronary, right-coronary and non-coronary sinuses.

Aortic arch

Main article: Aortic arch

The aortic arch loops over the left pulmonary artery and the bifurcation of the pulmonary trunk, to which it remains connected by the ligamentum arteriosum, a remnant of the fetal circulation that is obliterated a few days after birth. In addition to these blood vessels, the aortic arch crosses the left main bronchus. Between the aortic arch and the pulmonary trunk is a network of autonomic nerve fibers, the cardiac plexus or aortic plexus. The left vagus nerve, which passes anterior to the aortic arch, gives off a major branch, the recurrent laryngeal nerve, which loops under the aortic arch just lateral to the ligamentum arteriosum. It then runs back to the neck.

The aortic arch has three major branches: from proximal to distal, they are the brachiocephalic trunk, the left common carotid artery, and the left subclavian artery. The brachiocephalic trunk supplies the right side of the head and neck as well as the right arm and chest wall, while the latter two together supply the left side of the same regions.

The aortic arch ends, and the descending aorta begins at the level of the intervertebral disc between the fourth and fifth thoracic vertebrae.

Thoracic aorta

Main article: Thoracic aorta

The thoracic aorta gives rise to the intercostal and subcostal arteries, as well as to the superior and inferior left bronchial arteries and variable branches to the esophagus, mediastinum, and pericardium. Its lowest pair of branches are the superior phrenic arteries, which supply the diaphragm, and the subcostal arteries for the twelfth rib.

Abdominal aorta

Main article: Abdominal aorta

The abdominal aorta begins at the aortic hiatus of the diaphragm at the level of the twelfth thoracic vertebra. It gives rise to lumbar and musculophrenic arteries, renal and middle suprarenal arteries, and visceral arteries (the celiac trunk, the superior mesenteric artery and the inferior mesenteric artery). It ends in a bifurcation into the left and right common iliac arteries. At the point of the bifurcation, there also springs a smaller branch, the median sacral artery.

Development

The ascending aorta develops from the outflow tract, which initially starts as a single tube connecting the heart with the aortic arches (which will form the great arteries) in early development but is then separated into the aorta and the pulmonary trunk.

The aortic arches start as five pairs of symmetrical arteries connecting the heart with the dorsal aorta, and then undergo a significant remodelling to form the final asymmetrical structure of the great arteries, with the 3rd pair of arteries contributing to the common carotids, the right 4th forming the base and middle part of the right subclavian artery and the left 4th being the central part of the aortic arch. The smooth muscle of the great arteries and the population of cells that form the aorticopulmonary septum that separates the aorta and pulmonary artery is derived from cardiac neural crest. This contribution of the neural crest to the great artery smooth muscle is unusual as most smooth muscle is derived from mesoderm. In fact the smooth muscle within the abdominal aorta is derived from mesoderm, and the coronary arteries, which arise just above the semilunar valves, possess smooth muscle of mesodermal origin. A failure of the aorticopulmonary septum to divide the great vessels results in persistent truncus arteriosus.

Microanatomy

A pig's aorta cut open, also showing some branching arteries.

The aorta is an elastic artery, and as such is quite distensible. The aorta consists of a heterogeneous mixture of smooth muscle, nerves, intimal cells, endothelial cells, fibroblast-like cells, and a complex extracellular matrix. The vascular wall is subdivided into three layers known as the tunica externa, tunica media, and tunica intima. The aorta is covered by an extensive network of tiny blood vessels called vasa vasora, which feed the tunica externa and tunica media, the outer layers of the aorta. The aortic arch contains baroreceptors and chemoreceptors that relay information concerning blood pressure and blood pH and carbon dioxide levels to the medulla oblongata of the brain. This information along with information from baroreceptors and chemoreceptors located elsewhere is processed by the brain and the autonomic nervous system mediates appropriate homeostatic responses.

Within the tunica media, smooth muscle and the extracellular matrix are quantitatively the largest components, these are arranged concentrically as musculoelastic layers (the elastic lamella) in mammals. The elastic lamella, which comprise smooth muscle and elastic matrix, can be considered as the fundamental structural unit of the aorta and consist of elastic fibers, collagens (predominately type III), proteoglycans, and glycoaminoglycans. The elastic matrix dominates the biomechanical properties of the aorta. The smooth muscle component, while contractile, does not substantially alter the diameter of the aorta, but rather serves to increase the stiffness and viscoelasticity of the aortic wall when activated.

Variation

Variations may occur in the location of the aorta, and the way in which arteries branch off the aorta. The aorta, normally on the left side of the body, may be found on the right in dextrocardia, in which the heart is found on the right, or situs inversus, in which the location of all organs are flipped.

Variations in the branching of individual arteries may also occur. For example, the left vertebral artery may arise from the aorta, instead of the left common carotid artery.

In patent ductus arteriosus, a congenital disorder, the fetal ductus arteriosus fails to close, leaving an open vessel connecting the pulmonary artery to the proximal descending aorta.

Function

Major aorta anatomy displaying ascending aorta, brachiocephalic trunk, left common carotid artery, left subclavian artery, aortic isthmus, aortic arch, and descending thoracic aorta

The aorta supplies all of the systemic circulation, which means that the entire body, except for the respiratory zone of the lung, receives its blood from the aorta. Broadly speaking, branches from the ascending aorta supply the heart; branches from the aortic arch supply the head, neck, and arms; branches from the thoracic descending aorta supply the chest (excluding the heart and the respiratory zone of the lung); and branches from the abdominal aorta supply the abdomen. The pelvis and legs get their blood from the common iliac arteries.

Blood flow and velocity

The contraction of the heart during systole is responsible for ejection and creates a (pulse) wave that is propagated down the aorta, into the arterial tree. The wave is reflected at sites of impedance mismatching, such as bifurcations, where reflected waves rebound to return to semilunar valves and the origin of the aorta. These return waves create the dicrotic notch displayed in the aortic pressure curve during the cardiac cycle as these reflected waves push on the aortic semilunar valve. With age, the aorta stiffens such that the pulse wave is propagated faster and reflected waves return to the heart faster before the semilunar valve closes, which raises the blood pressure. The stiffness of the aorta is associated with a number of diseases and pathologies, and noninvasive measures of the pulse wave velocity are an independent indicator of hypertension. Measuring the pulse wave velocity (invasively and non-invasively) is a means of determining arterial stiffness. Maximum aortic velocity may be noted as V_max or less commonly as AoV_max.

Mean arterial pressure (MAP) is highest in the aorta, and the MAP decreases across the circulation from aorta to arteries to arterioles to capillaries to veins back to atrium. The difference between aortic and right atrial pressure accounts for blood flow in the circulation. When the left ventricle contracts to force blood into the aorta, the aorta expands. This stretching gives the potential energy that will help maintain blood pressure during diastole, as during this time the aorta contracts passively. This Windkessel effect of the great elastic arteries has important biomechanical implications. The elastic recoil helps conserve the energy from the pumping heart and smooth out the pulsatile nature created by the heart. Aortic pressure is highest at the aorta and becomes less pulsatile and lower pressure as blood vessels divide into arteries, arterioles, and capillaries such that flow is slow and smooth for gases and nutrient exchange.

Clinical significance

• Aortic aneurysm – mycotic, bacterial (e.g. syphilis), senile, genetic, associated with valvular heart disease
• Aortic coarctation – pre-ductal, post-ductal
• Aortic dissection
• Aortic stenosis
• Abdominal aortic aneurysm
• Aortitis, inflammation of the aorta that can be seen in trauma, infections, and autoimmune disease
• Atherosclerosis
• Ehlers–Danlos syndrome
• Marfan syndrome
• Trauma, such as traumatic aortic rupture, most often thoracic and distal to the left subclavian artery and often quickly fatal
• Transposition of the great vessels, see also dextro-Transposition of the great arteries and levo-Transposition of the great arteries

Other animals

All amniotes have a broadly similar arrangement to that of humans, albeit with a number of individual variations. In fish, however, there are two separate vessels referred to as aortas. The ventral aorta carries de-oxygenated blood from the heart to the gills; part of this vessel forms the ascending aorta in tetrapods (the remainder forms the pulmonary artery). A second, dorsal aorta carries oxygenated blood from the gills to the rest of the body and is homologous with the descending aorta of tetrapods. The two aortas are connected by a number of vessels, one passing through each of the gills. Amphibians also retain the fifth connecting vessel, so that the aorta has two parallel arches.

History

The word aorta stems from the Late Latin aorta from Classical Greek aortē (ἀορτή), from aeirō, "I lift, raise" (ἀείρω) This term was first applied by Aristotle when describing the aorta and describes accurately how it seems to be "suspended" above the heart.

The function of the aorta is documented in the Talmud, where it is noted as one of three major vessels entering or leaving the heart, and where perforation is linked to death.

Artery

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Artery

 

Artery
Diagram of an artery
Details
Identifiers
Latin	Arteria (plural: arteriae)
MeSH	D001158
TA98	A12.0.00.003 A12.2.00.001
TA2	3896
FMA	50720

An artery (PL: arteries) (from Greek ἀρτηρία (artēríā) 'windpipe, artery') is a blood vessel in humans and most other animals that takes blood away from the heart to one or more parts of the body (tissues, lungs, brain etc.). Most arteries carry oxygenated blood; the two exceptions are the pulmonary and the umbilical arteries, which carry deoxygenated blood to the organs that oxygenate it (lungs and placenta, respectively). The effective arterial blood volume is that extracellular fluid which fills the arterial system.

The arteries are part of the circulatory system, that is responsible for the delivery of oxygen and nutrients to all cells, as well as the removal of carbon dioxide and waste products, the maintenance of optimum blood pH, and the circulation of proteins and cells of the immune system.

Arteries contrast with veins, which carry blood back towards the heart.

Structure

Microscopic anatomy of an artery.

Cross-section of a human artery

See also: Arterial tree

The anatomy of arteries can be separated into gross anatomy, at the macroscopic level, and microanatomy, which must be studied with a microscope. The arterial system of the human body is divided into systemic arteries, carrying blood from the heart to the whole body, and pulmonary arteries, carrying deoxygenated blood from the heart to the lungs.

The outermost layer of an artery (or vein) is known as the tunica externa, also known as tunica adventitia, and is composed of collagen fibers and elastic tissue - with the largest arteries containing vasa vasorum (small blood vessels that supply large blood vessels). Most of the layers have a clear boundary between them, however the tunica externa has a boundary that is ill-defined. Normally its boundary is considered when it meets or touches the connective tissue. Inside this layer is the tunica media, or media, which is made up of smooth muscle cells, elastic tissue (also called connective tissue proper) and collagen fibres. The innermost layer, which is in direct contact with the flow of blood, is the tunica intima, commonly called the intima. The elastic tissue allows the artery to bend and fit through places in the body. This layer is mainly made up of endothelial cells (and a supporting layer of elastin rich collagen in elastic arteries). The hollow internal cavity in which the blood flows is called the lumen.

Development

Further information: Angiogenesis and Vasculogenesis

Arterial formation begins and ends when endothelial cells begin to express arterial specific genes, such as ephrin B2.

Function

Arteries form part of the human circulatory system

Main article: Circulatory system

Arteries form part of the circulatory system. They carry blood that is oxygenated after it has been pumped from the heart. Coronary arteries also aid the heart in pumping blood by sending oxygenated blood to the heart, allowing the muscles to function. Arteries carry oxygenated blood away from the heart to the tissues, except for pulmonary arteries, which carry blood to the lungs for oxygenation (usually veins carry deoxygenated blood to the heart but the pulmonary veins carry oxygenated blood as well). There are two types of unique arteries. The pulmonary artery carries blood from the heart to the lungs, where it receives oxygen. It is unique because the blood in it is not "oxygenated", as it has not yet passed through the lungs. The other unique artery is the umbilical artery, which carries deoxygenated blood from a fetus to its mother.

Arteries have a blood pressure higher than other parts of the circulatory system. The pressure in arteries varies during the cardiac cycle. It is highest when the heart contracts and lowest when heart relaxes. The variation in pressure produces a pulse, which can be felt in different areas of the body, such as the radial pulse. Arterioles have the greatest collective influence on both local blood flow and on overall blood pressure. They are the primary "adjustable nozzles" in the blood system, across which the greatest pressure drop occurs. The combination of heart output (cardiac output) and systemic vascular resistance, which refers to the collective resistance of all of the body's arterioles, are the principal determinants of arterial blood pressure at any given moment.

Arteries have the highest pressure and have narrow lumen diameter. It consists of three tunics: Tunica media, intima, and external.

Systemic arteries are the arteries (including the peripheral arteries), of the systemic circulation, which is the part of the cardiovascular system that carries oxygenated blood away from the heart, to the body, and returns deoxygenated blood back to the heart. Systemic arteries can be subdivided into two types—muscular and elastic—according to the relative compositions of elastic and muscle tissue in their tunica media as well as their size and the makeup of the internal and external elastic lamina. The larger arteries (>10  mm diameter) are generally elastic and the smaller ones (0.1–10 mm) tend to be muscular. Systemic arteries deliver blood to the arterioles, and then to the capillaries, where nutrients and gases are exchanged.

After traveling from the aorta, blood travels through peripheral arteries into smaller arteries called arterioles, and eventually to capillaries. Arterioles help in regulating blood pressure by the variable contraction of the smooth muscle of their walls, and deliver blood to the capillaries.

Aorta

Aorta is the largest blood vessel in human body

The aorta is the root systemic artery (i.e., main artery). In humans, it receives blood directly from the left ventricle of the heart via the aortic valve. As the aorta branches and these arteries branch, in turn, they become successively smaller in diameter, down to the arterioles. The arterioles supply capillaries, which in turn empty into venules. The first branches off of the aorta are the coronary arteries, which supply blood to the heart muscle itself. These are followed by the branches of the aortic arch, namely the brachiocephalic artery, the left common carotid, and the left subclavian arteries.

Capillaries

Main article: Capillaries

The capillaries are the smallest of the blood vessels and are part of the microcirculation. The microvessels have a width of a single cell in diameter to aid in the fast and easy diffusion of gases, sugars and nutrients to surrounding tissues. Capillaries have no smooth muscle surrounding them and have a diameter less than that of red blood cells; a red blood cell is typically 7 micrometers outside diameter, capillaries typically 5 micrometers inside diameter. The red blood cells must distort in order to pass through the capillaries.

These small diameters of the capillaries provide a relatively large surface area for the exchange of gases and nutrients.

Clinical significance

Diagram showing the effects of atherosclerosis on an artery.

Systemic arterial pressures are generated by the forceful contractions of the heart's left ventricle. High blood pressure is a factor in causing arterial damage. Healthy resting arterial pressures are relatively low, mean systemic pressures typically being under 100 mmHg (1.9 psi; 13 kPa) above surrounding atmospheric pressure (about 760 mmHg, 14.7 psi, 101 kPa at sea level). To withstand and adapt to the pressures within, arteries are surrounded by varying thicknesses of smooth muscle which have extensive elastic and inelastic connective tissues. The pulse pressure, being the difference between systolic and diastolic pressure, is determined primarily by the amount of blood ejected by each heart beat, stroke volume, versus the volume and elasticity of the major arteries.

A blood squirt also known as an arterial gush is the effect when an artery is cut due to the higher arterial pressures. Blood is spurted out at a rapid, intermittent rate, that coincides with the heartbeat. The amount of blood loss can be copious, can occur very rapidly, and be life-threatening.

Over time, factors such as elevated arterial blood sugar (particularly as seen in diabetes mellitus), lipoprotein, cholesterol, high blood pressure, stress and smoking, are all implicated in damaging both the endothelium and walls of the arteries, resulting in atherosclerosis. Atherosclerosis is a disease marked by the hardening of arteries. This is caused by an atheroma or plaque in the artery wall and is a build-up of cell debris, that contain lipids, (cholesterol and fatty acids), calcium and a variable amount of fibrous connective tissue.

Accidental intraarterial injection either iatrogenically or through recreational drug use can cause symptoms such as intense pain, paresthesia and necrosis. It usually causes permanent damage to the limb; often amputation is necessary.

History

Further information: Circulatory system § History

Among the Ancient Greeks before Hippocrates, all blood vessels were called Φλέβες, phlebes. The word arteria then referred to the windpipe. Herophilos was the first to describe anatomical differences between the two types of blood vessel. While Empedocles believed that the blood moved to and fro through the blood vessels, there was no concept of the capillary vessels that join arteries and veins, and there was no notion of circulation. Diogenes of Apollonia developed the theory of pneuma, originally meaning just air but soon identified with the soul itself, and thought to co-exist with the blood in the blood vessels. The arteries were thought to be responsible for the transport of air to the tissues and to be connected to the trachea. This was as a result of finding the arteries of cadavers devoid of blood.

In medieval times, it was supposed that arteries carried a fluid, called "spiritual blood" or "vital spirits", considered to be different from the contents of the veins. This theory went back to Galen. In the late medieval period, the trachea, and ligaments were also called "arteries".

William Harvey described and popularized the modern concept of the circulatory system and the roles of arteries and veins in the 17th century.

Alexis Carrel at the beginning of the 20th century first described the technique for vascular suturing and anastomosis and successfully performed many organ transplantations in animals; he thus actually opened the way to modern vascular surgery that was previously limited to vessels' permanent ligation.