title: Nayanthara

text: In 2008 she had five releases, four of which were in Tamil. Her first release was the family entertainer "Yaaradi Nee Mohini". Behindwoods.com wrote: "Nayan dispels her bombshell act and proves that she can more than just that. She breaks into tears when needed, shows vicious contempt when rubbed the wrong way in the name of love, and looks endearing in songs". Nowrunning.com stated that she "exhibits her acting skills in full measure [and] gives a moving performance in emotional scenes", while Sify's critic wrote that she looked "sensational and has done a great job in perhaps the meatiest role she has done so far". She won the Vijay Award for Favourite Heroine for the second consecutive year, and was nominated for the Best Actress prize at the 56th Filmfare Awards South.


title: Nayanthara

text: Her subsequent appearances were in "Kuselan", "Satyam" and "Aegan" in 2008. In 2009 she appeared in "Villu" and "Aadhavan".


title: Nayanthara

text: In 2010, all her releases, which featured her as the female lead, turned out to be commercial successes: she had five box office hits in the four Southern languages – "Adhurs" (Telugu), "Bodyguard" (Malayalam), "Simha" (Telugu), "Boss Engira Bhaskaran" (Tamil) and "Super" (Kannada). She won the Asianet Award for Best Actress for her performance in "Bodyguard". The latter three, in particular, were particularly good for Nayanthara, with "Simha" becoming one of the highest-grossing Telugu films of the year and "Boss Engira Bhaskaran" releasing to positive reviews, and becoming a financial success. Upendra's "Super", which marked her debut into the Kannada film industry, opened to rave reviews, while her performance was also praised by critics. Her performances in "Simha", "Boss Engira Bhaskaran" and "Super" eventually fetched her nominations for the Filmfare Best Actress Award in the respective languages. She also starred in Shyamaprasad's critically acclaimed Malayalam film "Elektra". Her performance was well appreciated by critics when it was screened at the International Film Festival of India. The film was also screened at the Dubai International Film Festival.


title: Nayanthara

text: Her only release in 2011 was the mythological film "Sri Rama Rajyam" (2011) by Bapu, in which she played the role of Sita. She received critical acclaim for her performance in the film, with Rediff.com noting, "Nayanthara is the surprise package of the film. As Sita, she too has played the role of her lifetime. She gave a fine understated performance conveying a kaleidoscope of emotions." Sify commented "Nayanthara gave a fitting answer to all, unleashing grace and accomplishing the assignment with absolute perfection." Subsequently, she was awarded the Nandi Award for Best Actress and the Filmfare Award for Best Telugu Actress. The film was dubbed in the media as her swan song and considered to be her last release before her marriage as she was reported to quit acting thereafter.


title: Nayanthara

text: Choosing to make a comeback, Nayanthara began shooting for Krish's "Krishnam Vande Jagadgurum" (2012) after an eleven-month sabbatical. She portrayed the role of a journalist and her performance was praised, with a reviewer from Sify stating that "Nayanthara does not play the normal commercial glam-doll for a change and she is good" and "her chemistry with Rana is perfect". The following year, she won critical acclaim for her performance in Atlee's romantic drama "Raja Rani" (2013), winning both the Filmfare and Vijay Award for Best Actress. Playing the role of a wife in an unhappy marriage, a critic from Sify.com noted, "Nayanthara gets a well-written role and she couldn't ask for a better character than Regina for her return after a three-year hiatus... she looks good, throws herself into the role, approaches her part with restraint and minimalist fuss, and turns in a sincere performance." Sify.com also said her role was "unquestionably the film's biggest strength". Likewise, another reviewer from Behindwoods.com stated the role was "Nayanthara's career best performance till date" and it was a "great comeback by the actress with a role which gave her ample scope".


title: Nayanthara

text: Her next release, Vishnuvardhan's "Arrambam" (2013), featuring Ajith Kumar in the lead, also won acclaim with a reviewer from Rediff stating that "Nayanthara looks gorgeous and does full justice to her character", while Behindwoods critic stated that "Nayanthara puts out a fiery face when required and also showcase her oomph and style in many scenes, adding "this superstar continues to shine".


title: Nayanthara

text: Nayanthara's rising popularity as an actress led to her being cast in female-centric films, a genre considered a rarity in South Indian cinema. Sekhar Kammula cast her in the titular role of his bilingual "Anaamika" (2014), a remake of the Hindi film "Kahaani" (2012), in which Nayanthara portrayed an IT professional who moves to Hyderabad to look for her missing husband. Sify.com labelled her performance in the film "a one-woman show by her in a difficult role... she nails the character perfectly whether it in the initial scenes showing her anguish or in the stunning climax, where she takes a brave decision". In contrast, Rediff.com stated that her "performance is lacklustre in the first half".


title: Nayanthara

text: While "Anaamika" did middling business, her next female-centric project, the horror film "Maya" (2015), emerged as a profitable venture commercially. Portraying a single mother with a mysterious past, Nayanthara won acclaim for her performance and her decision to portray an unconventional character, with Sify.com saying, "Nayanthara is just fabulous and her classier side has been perfectly unleashed in "Maya"... it's a treat to watch Nayanthara re-inventing herself by performing in an author backed role".


title: Nayanthara

text: Along with strong character roles, Nayanthara continued to appear in action and comedy films, including the successful "Bhaskar the Rascal" (2015) and "Thani Oruvan" (2015), during the same period. She then went on to win her second Filmfare Award for her role of a deaf girl seeking revenge in Vignesh Shivan's black comedy "Naanum Rowdy Dhaan" (2015). About her performance, Sify.com's reviewer noted "Nayanthara's Kadambari is a well-etched out role, the actress is effortlessly charming and she does full justice", and "infuses the part with the right portions of vulnerability, restraint, and quiet strength, delivering a performance that is nothing short of perfect". A video clip from the movie "Naanum Rowdy Dhaan" went viral on social media where Nayanthara was seen buying liquor from a shop. The Hindu Makkal Katchi demanded the scene to be removed and garlanded her picture with beer bottles and even after she made a statement that it was a film scene, the Hindu Munnani burned her effigy in protest.


title: Nayanthara

text: In her first release of 2016, Nayanthara portrayed a rape victim in "Puthiya Niyamam" (2016) opposite Mammootty, with a critic from the IBTimes stating "Nayantara's role as Vasuki Iyer is one of her career's best until now and her decision to dub in her own voice for her character is worthy of appreciation". She continued to play diverse roles, appearing as a village-based school teacher in "Thirunaal" (2016), a spy in the commercially successful Vikram-starrer "Iru Mugan" (2016) and the queen of a fantasy kingdom in "Kaashmora" (2016).


title: Nayanthara

text: In 2017, she played the main lead in the horror thriller "Dora" and the critically acclaimed social drama "Aramm". Her portrayal of a district collector in "Aramm" won her the third Filmfare Award for Best Tamil Actress. She also played the female lead in the action thriller "Velaikkaran". Her first release in 2018 was a Telugu-language film "Jai Simha" in which she played the female lead. She went onto portray the main lead in Nelson Dilipkumar directorial "Kolamaavu Kokila" and the thriller "Imaikkaa Nodigal" alongside actor Atharvaa.


title: Nayanthara

text: In 2019, she had lead roles in "Viswasam, Airaa, Kolaiyuthir Kaalam and Bigil", the latter in which she collaborated with Vijay for the third time. She also appeared in the Malayalam film "Love Action Drama", alongside Nivin Pauly and the Telugu biographical film, "Sye Raa Narasimha Reddy", alongside Chiranjeevi and Tamannaah. In 2020, she was seen in a supporting role in "Darbar" along with Rajinikanth. Her film "Mookuthi Amman" marked her second entry into Hindu mythological and devotional cinema after her portrayal of Goddess Sita in 2011.


title: Nayanthara

text: 2021 saw her fifth collaboration with Rajinikanth with the film "Annaatthe" where she was seen as a lawyer. She went into star in "Netrikann", a remake of the 2011 South Korean film "Blind". She also starred in the Malayalam film "Nizhal" and the much delayed Telugu film "Aaradugula Bullet".


title: Nayanthara

text: Her first release of 2022 was the romcom-drama "Kaathuvaakula Rendu Kaadhal", where she was seen alongside Vijay Sethupathi and Samantha Ruth Prabhu. She later starred in G.S. Viknesh's survival-thriller "O2", where she played a single mother trying to shield her son suffering from cystic fibrosis from the effects of low oxygen.


title: Nayanthara

text: She will be next seen in the Malayalam film "Gold", alongside Prithviraj Sukumaran, in the Telugu film "Godfather", the Tamil films "Connect" and "Iraivan" and the Bollywood film "Jawan" opposite Shah Rukh Khan which will mark her Hindi debut. She is also set to star in an untitled film, tentatively titled as "Lady Superstar 75".


title: Nayanthara

text: She was brought up as a Christian to Malayali parents. On 7 August 2011, she embraced Hinduism at the Arya Samaj Temple in Chennai. After that, a certificate of conversion to Hinduism was issued to her, and her stage name, "Nayanthara", became her official name. She is a polydactyl, with a rudimentary finger on her left hand.


title: Nayanthara

text: Nayanthara acknowledged her three and a half-year live-in relationship with actor and dance choreographer Prabhu Deva.


title: Nayanthara

text: In 2015, Nayanthara and director Vignesh Shivan have been in a relationship ever since they worked together on "Naanum Rowdy Dhaan" in 2015. The couple got married on 9 June 2022 in Mahabalipuram.


title: Nayanthara

text: In 2021, Nayanthara and Vignesh Shivan launched their film production venture Rowdy Pictures, which produced the films "Koozhangal", "Netrikann" and "Kaathuvaakula Rendu Kaadhal". In the same year, Nayanthara, along with renowned dermatologist Dr. Renita Rajan, launched The Lip Balm Company.


title: Prime number

text: 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, or , involve 5 itself.


title: Prime number

text: 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.


title: Prime number

text: The property of being prime is called primality. A simple but slow method of checking the primality of a given number formula_1, called trial division, tests whether formula_1 is a multiple of any integer between 2 and formula_3. 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. the largest known prime number is a Mersenne prime with 24,862,048 decimal digits.


title: Prime number

text: 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.


title: Prime number

text: 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 having just one even number between them. 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.


title: Prime number

text: 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, formula_1 is prime if formula_1 items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange formula_1 dots into a rectangular grid that is more than one dot wide and more than one dot high.


title: Prime number

text: 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).


title: Prime number

text: Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This idea leads to a different but equivalent definition of the primes: they are the numbers with exactly two positive divisors, 1 and the number itself.


title: Prime number

text: Yet another way to express the same thing is that a number formula_1 is prime if it is greater than one and if none of the numbers formula_10 divides formula_1 evenly.


title: Prime number

text: No even number formula_1 greater than 2 is prime because any such number can be expressed as the product formula_13. 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:


title: Prime number

text: 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.


title: Prime number

text: The set of all primes is sometimes denoted by formula_14 (a boldface capital "P") or by formula_15 (a blackboard bold capital P).


title: Prime number

text: 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


title: Prime number

text: Around 1000 AD, the Islamic mathematician Ibn al-Haytham (Alhazen) found Wilson's theorem, characterizing the prime numbers as the numbers formula_1 that evenly divide formula_17. 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 brought the innovations from Islamic mathematics back 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.


title: Prime number

text: In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Fermat also investigated the primality of the and Marin Mersenne studied the Mersenne primes, prime numbers of the form formula_18 with formula_19 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 formula_20.


title: Prime number

text: At the start of the 19th century, Legendre and Gauss conjectured that as formula_21 tends to infinity, the number of primes up to formula_21 is asymptotic to formula_23, where formula_24 is the natural logarithm of formula_21. A weaker consequence of this high density of primes was Bertrand's postulate, that for every formula_26 there is a prime between formula_1 and formula_28, 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.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: 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".


title: Prime number

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


title: Prime number

text: 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 formula_30 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, formula_31 denotes the square or second power of formula_30


title: Prime number

text: 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,


title: Prime number

text: 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,


title: Prime number

text: 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.


title: Prime number

text: Some proofs of the uniqueness of prime factorizations are based on Euclid's lemma: If formula_19 is a prime number and formula_19 divides a product formula_35 of integers formula_36 and formula_37 then formula_19 divides formula_36 or formula_19 divides formula_41 (or both). Conversely, if a number formula_19 has the property that when it divides a product it always divides at least one factor of the product, then formula_19 must be prime.


title: Prime number

text: 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.


title: Prime number

text: 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 formula_44 If the list consists of the primes formula_45 this gives the number


title: Prime number

text: with one or more prime factors. formula_47 is evenly divisible by each of these factors, but formula_47 has a remainder of one when divided by any of the prime numbers in the given list, so none of the prime factors of formula_47 can be in the given list. Because there is no finite list of all the primes, there must be infinitely many primes.


title: Prime number

text: 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,


title: Prime number

text: 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.


title: Prime number

text: Other examples of prime-generating formulas come from Mills' theorem and a theorem of Wright. These assert that there are real constants formula_53 and formula_54 such that


title: Prime number

text: are prime for any natural number formula_1 in the first formula, and any number of exponents in the second formula. Here formula_57 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 formula_58 or formula_59


title: Prime number

text: 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 formula_1 greater than 2 can be written as a sum of two primes. , this conjecture has been verified for all numbers up to formula_61 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.


title: Prime number

text: The existence of arbitrarily large prime gaps can be seen by noting that the sequence formula_62 consists of formula_63 composite numbers, for any natural number formula_64 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 formula_65 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 formula_66 there are infinitely many pairs of consecutive primes that differ by formula_67


title: Prime number

text: Andrica's conjecture, Brocard's conjecture, Legendre's conjecture, and Oppermann's conjecture all suggest that the largest gaps between primes from formula_68 to formula_1 should be at most approximately formula_70 a result that is known to follow from the Riemann hypothesis, while the much stronger Cramér conjecture sets the largest gap size at formula_71 Prime gaps can be generalized to prime formula_72-tuples, patterns in the differences between 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.


title: Prime number

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


title: Prime number

text: This area of study began with Leonhard Euler and his first major result, the solution to the Basel problem.


title: Prime number

text: which today can be recognized as the value formula_74 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 formula_75.


title: Prime number

text: The reciprocal of this number, formula_76, is the limiting probability that two random numbers selected uniformly from a large range are relatively prime (have no factors in common).


title: Prime number

text: 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 formula_1-th prime is known.


title: Prime number

text: with relatively prime integers formula_36 and formula_41 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 formula_41 have approximately the same proportions of primes.


title: Prime number

text: 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.


title: Prime number

text: Euler showed that, for any arbitrary real number formula_21, there exists a prime formula_19 for which this sum is bigger than formula_21. 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 formula_21.


title: Prime number

text: The growth rate of this sum is described more precisely by Mertens' second theorem. For comparison, the sum


title: Prime number

text: does not grow to infinity as formula_1 goes to infinity (see the Basel problem). In this sense, prime numbers occur more often than squares of natural numbers,


title: Prime number

text: although both sets are infinite. Brun's theorem states that the sum of the reciprocals of twin primes,


title: Prime number

text: 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.


title: Prime number

text: The prime-counting function formula_90 is defined as the number of primes not greater than formula_1. For example, formula_92, since there are five primes less than or equal to 11. Methods such as the Meissel–Lehmer algorithm can compute exact values of formula_90 faster than it would be possible to list each prime up to formula_1. The prime number theorem states that formula_90 is asymptotic to formula_96, which is denoted as


title: Prime number

text: and means that the ratio of formula_90 to the right-hand fraction approaches 1 as formula_1 grows to infinity. This implies that the likelihood that a randomly chosen number less than formula_1 is prime is (approximately) inversely proportional to the number of digits in formula_1.


title: Prime number

text: 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,


title: Prime number

text: 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


title: Prime number

text: can have more than one prime only when its remainder formula_36 and modulus formula_109 are relatively prime. If they are relatively prime, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes.


title: Prime number

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


title: Prime number

text: yields prime numbers for formula_111, 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


title: Prime number

text: in terms of the logarithmic integral and the polynomial coefficients. No quadratic polynomial has been proven to take infinitely many prime values.


title: Prime number

text: 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.


title: Prime number

text: 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 formula_112 are located.


title: Prime number

text: This function is an analytic function on the complex numbers. For complex numbers formula_113 with real part greater than one it equals both an infinite sum over all integers, and an infinite product over the prime numbers,


title: Prime number

text: 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.


title: Prime number

text: then the sum-product equality would also be valid at formula_115, but the sum would diverge (it is the harmonic series formula_116) while the product would be finite, a contradiction.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: 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


title: Prime number

text: asymptotic distribution of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of formula_21 for intervals near a number formula_21).


title: Prime number

text: Modular arithmetic modifies usual arithmetic by only using the numbers formula_119, for a natural number formula_1 called the modulus.


title: Prime number

text: Any other natural number can be mapped into this system by replacing it by its remainder after division by formula_1.


title: Prime number

text: Modular sums, differences and products are calculated by performing the same replacement by the remainder


title: Prime number

text: on the result of the usual sum, difference, or product of integers. Equality of integers corresponds to "congruence" in modular arithmetic:


title: Prime number

text: formula_21 and formula_123 are congruent (written formula_124 mod formula_1) when they have the same remainder after division by formula_1. 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 formula_127 as modulus, division by formula_128 is possible: formula_129, because clearing denominators by multiplying both sides by formula_128 gives the valid formula formula_131. However, with the composite modulus formula_132, division by formula_128 is impossible. There is no valid solution to formula_134: clearing denominators by multiplying by formula_128 causes the left-hand side to become formula_136 while the right-hand side becomes either formula_137 or formula_128.


title: Prime number

text: 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.


title: Prime number

text: Several theorems about primes can be formulated using modular arithmetic. For instance, Fermat's little theorem states that if


title: Prime number

text: Giuga's conjecture says that this equation is also a sufficient condition for formula_19 to be prime.


title: Prime number

text: Wilson's theorem says that an integer formula_147 is prime if and only if the factorial formula_148 is congruent to formula_149 mod formula_19. For a composite this cannot hold, since one of its factors divides both and formula_151, and so formula_152 is impossible.


title: Prime number

text: The formula_19-adic order formula_154 of an integer formula_1 is the number of copies of formula_19 in the prime factorization of formula_1. The same concept can be extended from integers to rational numbers by defining the formula_19-adic order of a fraction formula_159 to be formula_160. The formula_19-adic absolute value formula_162 of any rational number formula_109 is then defined as


title: Prime number

text: formula_164. Multiplying an integer by its formula_19-adic absolute value cancels out the factors of formula_19 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 formula_19-adic distance, the formula_19-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 formula_19. 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 formula_19-adic distance can be extended to a different complete field, the formula_19-adic numbers.


title: Prime number

text: 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 formula_19-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.


title: Prime number

text: 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 formula_19 of a ring formula_174 is called prime if it is nonzero, has no multiplicative inverse (that is, it is not a unit), and satisfies the following requirement: whenever formula_19 divides the product formula_176 of two elements of formula_174, it also divides at least one of formula_21 or formula_123. 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,


title: Prime number

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


title: Prime number

text: The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example of such a domain is the Gaussian integers formula_181, the ring of complex numbers of the form formula_182 where formula_183 denotes the imaginary unit and formula_36 and formula_41 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 formula_186 and formula_187. 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,


title: Prime number

text: which states that an odd prime formula_19 is expressible as the sum of two squares, formula_189, and therefore factorable as formula_190, exactly when formula_19 is 1 mod 4.


title: Prime number

text: Not every ring is a unique factorization domain. For instance, in the ring of numbers formula_192 (for integers formula_36 and formula_41) the number formula_195 has two factorizations formula_196, 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.


title: Prime number

text: "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.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: In the theory of finite groups the Sylow theorems imply that, if a power of a prime number formula_197 divides the order of a group, then the group has a subgroup of order formula_197. By Lagrange's theorem, any group of prime order is a cyclic group,


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: Prime numbers are also used in computing for checksums, hash tables, and pseudorandom number generators.


title: Prime number

text: The most basic method of checking the primality of a given integer formula_1 is called "trial division". This method divides formula_1 by each integer from 2 up to the square root of formula_1. Any such integer dividing formula_1 evenly establishes formula_1 as composite; otherwise it is prime.


title: Prime number

text: Integers larger than the square root do not need to be checked because, whenever formula_204, one of the two factors formula_36 and formula_41 is less than or equal to the square root of formula_1. Another optimization is to check only primes as factors in this range.


title: Prime number

text: For instance, to check whether 37 is prime, this method divides it by the primes in the range from 2 to formula_208, which are 2, 3, and 5. Each division produces a nonzero remainder, so 37 is indeed prime.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: Some of the fastest modern tests for whether an arbitrary given number formula_1 is prime are probabilistic (or Monte Carlo) algorithms, meaning that they have a small random chance of producing an incorrect answer.


title: Prime number

text: For instance the Solovay–Strassen primality test on a given number formula_19 chooses a number formula_36 randomly from formula_136 through formula_213 and uses modular exponentiation to check


title: Prime number

text: whether formula_214 is divisible by formula_19. If so, it answers yes and otherwise it answers no. If formula_19 really is prime, it will always answer yes, but if formula_19 is composite then it answers yes with probability at most 1/2 and no with probability at least 1/2.


title: Prime number

text: the probability that a composite number could pass the test every time is at most formula_219. 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.


title: Prime number

text: 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.


title: Prime number

text: The algorithms with guaranteed-correct output include both deterministic (non-random) algorithms, such as the AKS primality test,


title: Prime number

text: 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.


title: Prime number

text: When the elliptic curve method concludes that a number is prime, it provides primality certificate that can be verified quickly.


title: Prime number

text: 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;


title: Prime number

text: 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.


title: Prime number

text: The following table lists some of these tests. Their running time is given in terms of formula_1, the number to be tested and, for probabilistic algorithms, the number formula_72 of tests performed. Moreover, formula_222 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 formula_1 and formula_72.


title: Prime number

text: 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.


title: Prime number

text: For example, the Lucas–Lehmer primality test can determine whether a Mersenne number (one less than a power of two) is prime, deterministically,


title: Prime number

text: in the same time as a single iteration of the Miller–Rabin test. This is why since 1992 () the largest "known" prime has always been a Mersenne prime.


title: Prime number

text: 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.


title: Prime number

text: Given a composite integer formula_1, the task of providing one (or all) prime factors is referred to as "factorization" of formula_1. 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 formula_1, and elliptic curve factorization can be effective when formula_1 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. 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.


title: Prime number

text: 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. the largest number that has been factored by a quantum computer running Shor's algorithm is 21.


title: Prime number

text: 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 formula_21 and formula_123 than to calculate formula_21 and formula_123 (assumed coprime) if only the product formula_176 is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation (computing formula_234), while the reverse operation (the discrete logarithm) is thought to be a hard problem.


title: Prime number

text: 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 formula_72-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.


title: Prime number

text: 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 formula_236.


title: Prime number

text: Prime numbers are also used in pseudorandom number generators including linear congruential generators and the Mersenne Twister.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: with formula_72 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 formula_239 is composite and so are all other Fermat numbers that have been verified as of 2017. A regular formula_1-gon is constructible using straightedge and compass if and only if the odd prime factors of formula_1 (if any) are distinct Fermat primes. Likewise, a regular formula_1-gon may be constructed using straightedge, compass, and an angle trisector if and only if the prime factors of formula_1 are any number of copies of 2 or 3 together with a (possibly empty) set of distinct Pierpont primes, primes of the form formula_244.


title: Prime number

text: It is possible to partition any convex polygon into formula_1 smaller convex polygons of equal area and equal perimeter, when formula_1 is a power of a prime number, but this is not known for other values of formula_1.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: 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.


title: Prime number

text: 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".


title: Prime number

text: 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.


title: Mercedes-Benz

text: Mercedes-Benz (), commonly referred to as Mercedes and sometimes as Benz, is a German luxury and commercial vehicle automotive brand established in 1926. Mercedes-Benz AG (a Mercedes-Benz Group subsidiary established in 2019) is headquartered in Stuttgart, Baden-Württemberg, Germany. Mercedes-Benz AG produces consumer luxury vehicles and commercial vehicles badged as Mercedes-Benz. From November 2019 onwards, Mercedes-Benz-badged heavy commercial vehicles (trucks and buses) are managed by Daimler Truck, a former part of the Mercedes-Benz Group turned into an independent company in late 2021. In 2018, Mercedes-Benz was the largest brand of premium vehicles in the world, having sold 2.31 million passenger cars.


title: Mercedes-Benz

text: The brand's origins lie in Daimler-Motoren-Gesellschaft's 1901 Mercedes and Carl Benz's 1886 Benz Patent-Motorwagen, which is widely regarded as the first internal combustion engine in a self-propelled automobile. The slogan for the brand is "the best or nothing".


title: Mercedes-Benz

text: Mercedes-Benz traces its origins to Karl Benz's creation of the first internal combustion engine in a car, seen in the Benz Patent Motorwagen – financed by Bertha Benz's dowry and patented in January 1886 – and Gottlieb Daimler and their engineer Wilhelm Maybach's conversion of a stagecoach, with the addition of a petrol engine, introduced later that year. The Mercedes automobile was first marketed in 1901 by Daimler Motoren Gesellschaft (DMG).


title: Mercedes-Benz

text: Emil Jellinek, a European automobile entrepreneur who worked with DMG, registered the trademark in 1902, naming the 1901 Mercedes 35 hp after his daughter Mercedes Jellinek. Jellinek was a businessman and marketing strategist who promoted "horseless" Daimler automobiles among the highest circles of society in his adopted home. At the time, it was a meeting place for the "haute volée" of France and Europe, especially in winter. His customers included the Rothschild family and other well-known people, but Jellinek's plans went further, and as early as 1901, he was selling Mercedes cars in the "New World", as well, including United States billionaires Rockefeller, Astor, Morgan, and Taylor. At the Nice race he attended in 1899, Jellinek drove under the pseudonym "Monsieur Mercédès" as a way of concealing his less fancy real name. Many consider that race the birth of Mercedes-Benz as a brand. In 1901, the name "Mercedes" was re-registered by DMG worldwide as a protected trademark. The first Mercedes-Benz branded vehicles were produced in 1926, following the merger of Karl Benz and Gottlieb Daimler's companies into the Daimler-Benz company on 28 June of the same year.


title: Mercedes-Benz

text: Gottlieb Daimler was born on 17 March 1834 in Schorndorf. After training as a gunsmith and working in France, he attended the Polytechnic School in Stuttgart from 1857 to 1859. After completing various technical activities in France and England, he started working as a draftsman in Geislingen in 1862. At the end of 1863, he was appointed workshop inspector at a machine-tool factory in Reutlingen, where he met Wilhelm Maybach in 1865.


title: Mercedes-Benz

text: Throughout the 1930s, Mercedes-Benz produced the 770 model, a car that was notably popular throughout Germany's Nazi period. Adolf Hitler was known to have driven in a model of this car during his time in power, with modified custom bulletproof windshields. Most of the currently surviving 770 models were sold at auctions to private buyers. One of the cars is currently on display at the War Museum in Ottawa, Ontario. The pontiff's Popemobile has often been sourced from Mercedes-Benz.


title: Mercedes-Benz

text: From 1937 onward, Daimler Benz focused increasingly on military products such as the LG3000 lorry and the DB600 and the DB601 aero engines. To build the latter, in 1936, it built a factory hidden in the forest at Genshagen around 10 km south of Berlin. By 1942, the company had mostly stopped producing cars, and was now devoted to war production. According to its statement, in 1944, almost half of its 63,610 employees were forced labourers, prisoners of war, or concentration-camp detainees. Another source quotes this figure at 46,000. The company later paid $12 million in reparations to the labourers' families.


title: Mercedes-Benz

text: In 1958, the two companies began a partnership to sell their cars in the United States with Studebaker. A few American-based Daimler Benz dealerships were converted into Mercedes-Benz dealerships when Daimler's non-Mercedes-partnered company closed in 1966.


title: Mercedes-Benz

text: Over the decades, Mercedes-Benz has introduced many electronic and mechanical innovations and safety features that later became common. Currently, Mercedes-Benz is one of the best-known and longest-standing automotive brands in the world.


title: Mercedes-Benz

text: In November 2019, Daimler AG announced that Mercedes-Benz, until that point a company marque, would be spun off into a separate, wholly owned subsidiary called Mercedes-Benz AG. The new subsidiary would manage the Mercedes-Benz car and van business. Mercedes-Benz-badged trucks and buses would be part of the Daimler Truck AG subsidiary.


title: Mercedes-Benz

text: For information relating to the three-pointed star symbol of the brand, see under the title Daimler-Motoren-Gesellschaft, including the merger into Daimler-Benz.


title: Mercedes-Benz

text: In May 2022, Mercedes-Benz announced that it has recently sold the most expensive car at the price of $142 million. The car is a very rare 1955 Mercedes-Benz SLR that has been kept in the German automaker's collection and bought by a private owner. Mercedes in an announcement said that the sale will be used to establish the Mercedes-Benz Fund.


title: Mercedes-Benz

text: In June 2022, Mercedes-Benz recalled almost one million vehicles built between 2004 and 2015, due to potential problems with their braking system, caused by possible "advanced corrosion".


title: Mercedes-Benz

text: As of the Daimler AG company split, the Mercedes-Benz Cars division now handled the Mercedes-Benz- and the Smart-branded cars' production.


title: Mercedes-Benz

text: Mercedes-AMG became a majority-owned division of Mercedes-Benz in 1999. The company was integrated into DaimlerChrysler in 1999, and became Mercedes-Benz AMG on 1 January 1999.


title: Mercedes-Benz

text: Daimler's ultraluxury Maybach brand was under the Mercedes-Benz Cars division until December 2012, when production was stopped due to decreased sales. It now exists under the Mercedes-Maybach name, with the models being luxury-focused, enhanced models of Mercedes-Benz cars, such as the 2016 Mercedes-Maybach S600. The Mercedes-Maybach GLS 600 SUV debuted in November 2019.


title: Mercedes-Benz

text: Daimler partnered with BYD Auto to make and sell a battery-electric car called Denza in China. In 2016, Daimler announced plans to sell Mercedes-Benz-badged fully electric battery cars in China. Beijing Benz is a joint venture with the BAIC Group to produce the Mercedes-Benz branded cars in China. In 2018, Mercedes-Benz voluntarily apologized for sparking controversy within China by quoting the Dalai Lama on one of their promotional Instagram posts.


title: Mercedes-Benz

text: Other than in its native birthplace, Germany, Mercedes-Benz vehicles are or have been partly manufactured or assembled in:


title: Mercedes-Benz

text: Mercedes-Benz normally has a strong reputation for quality and durability. Some objective measures looking at passenger vehicles, such as J. D. Power surveys, demonstrated a downturn in reputation in these criteria in the late 1990s and early 2000s. By mid-2005, Mercedes temporarily returned to the industry average for initial quality, a measure of problems after the first 90 days of ownership, according to J. D. Power. In J. D. Power's Initial Quality Study for the first quarter of 2007, Mercedes showed dramatic improvement by climbing from 25th to 5th place and earning several awards for its newer models. For 2008, Mercedes-Benz's initial quality rating improved by yet another mark, to 4th place. On top of this accolade, it also received the Platinum Plant Quality Award for their Mercedes branded Sindelfingen, Germany body assembly plant. J. D. Power's 2011 US Initial Quality and Vehicle Dependability Studies both ranked Mercedes-Benz vehicles above average in build quality and reliability. In J. D. Power's United Kingdom Survey in 2011, Mercedes cars were rated above average. Additionally, iSeeCars.com study for Reuters in 2014 found Mercedes to have the lowest vehicle recall rate out of competitors.


title: Mercedes-Benz

text: Mercedes-Benz offers a versatile range of consumer-passenger, light commercial and heavy commercial equipment. These vehicles are manufactured in multiple countries worldwide. The Smart marque of city cars are also produced by Daimler AG.


title: Mercedes-Benz

text: Mercedes-Benz currently offers three types of vans; Citan, Vito, and Sprinter. They are all produced by Daimler AG.


title: Mercedes-Benz

text: Since December 2021, the Mercedes-Benz Trucks division is part of the Daimler Truck company and includes other sub-companies that were part of the DaimlerChrysler merger. Gottlieb Daimler sold the world's first truck in 1886. Their first factory to be built outside Germany after World War II was in Argentina. They originally built trucks, many of which were modified by third parties to be used as buses, popularly named Colectivo.


title: Mercedes-Benz

text: Mercedes-Benz has been producing buses since 1895 in Mannheim, Germany. Since 1995, the brands of Mercedes-Benz's buses and coaches are under the umbrella of EvoBus GmbH, since December 2021 belonging to Daimler Truck AG. EvoBus, through its regional subsidiaries, markets them in European countries, while in other regions of the world marketing and sales duties are passed to regional subsidiaries of Daimler Truck. Mercedes-Benz produces a wide range of buses and coaches, mainly for Europe and Asia. The first model was produced by Karl Benz in 1895.


title: Mercedes-Benz

text: The Mercedes-Benz 600 or 600S Pullman Guard limousines offer the option of armour-plating and have been used by diplomats worldwide.


title: Mercedes-Benz

text: Until 1994, Mercedes-Benz utilized an alphanumeric system for categorizing their vehicles, consisting of a number sequence approximately equal to the engine's displacement in liters multiplied by 100, followed by an arrangement of alphabetical suffixes, indicating body style and engine type.


title: Mercedes-Benz

text: Some models in the 1950s also had lower-case letters (b, c, and d) to indicate specific trim levels. For other models, the numeric part of the designation does not match the engine displacement. This was done to show the model's position in the model range independent of displacement or in the price matrix. For these vehicles, the actual displacement in litres is suffixed to the model designation. An exception was the 190-class with the numeric designation of "190" as to denote its entry-level in the model along with the displacement label on the right side of the boot (190E 2.3 for 2.3-litre 4-cylinder petrol motor, 190D 2.5 for 2.5-litre 5-cylinder diesel motor, and so forth). Some older models (such as the SS and SSK) did not have a number as part of the designation at all.


title: Mercedes-Benz

text: For the 1994 model year, Mercedes-Benz revised the naming system. Models were divided into "classes" denoted by an arrangement of up to three letters (see "Current model range" above), followed by a three-digit (or two-digit for AMG models, with the number approximately equal to the displacement in litres multiplied by 10) number related to the engine displacement as before. Variants of the same model such as an estate version or a vehicle with a diesel engine are no longer given a separate letter. The SLR and SLS supercars do not carry a numerical designation.


title: Mercedes-Benz

text: Today, many numerical designations no longer reflect the engine's actual displacement, but more of the relative performance and marketing position. Despite its engine displacement in two litres, the powerplant in the A45 AMG produces 355 brake horsepower so the designation is higher as to indicate the greater performance. Another example is the E250 CGI having greater performance than the E200 CGI due to the different engine tuning even though both have 1.8-litre engines. From the marketing perspective, E200 seems more "upscale" than E180. Recent AMG models use the "63" designation (in honor of the 1960s 6.3-litre M100 engine) despite being equipped with either a 6.2-litre (M156), a 5.5-litre (M157) or even a 4.0-litre engine.


title: Mercedes-Benz

text: Rationalisation of the model nomenclature was announced in November 2014 for future models. The changes consolidate many confusing nomenclature and their placements in the model range such as CL-Class is now called the S-Class Coupé. The naming structure is divided into four categories: core, off-road vehicle/SUV, 4-door coupé, and roadster. AMG GT and V-Class are unaffected by the change. In October 2016, Mercedes unveiled the X-Class; a pickup truck built on the Nissan Navara. At the 2016 Paris Motor Show, the company announced the EQ, a family of upcoming battery electric vehicles based on a modular platform, expected to represent up to 25% of its global sales by 2025.


title: Mercedes-Benz

text: Note: The CLA is positioned between the A- and B-Class models, while the CLS sits between the E- and S-Classes.


title: Mercedes-Benz

text: In addition to the revised nomenclature, Mercedes-Benz has new suffix nomenclature for the drive systems.


title: Mercedes-Benz

text: The revised A45 AMG for 2016 model year on has shifted the model designation to the right side while AMG is on the left side. This trend commenced with Mercedes-Maybach with MAYBACH on the left and S500/S600 on the right.


title: Mercedes-Benz

text: Mercedes-Benz has developed multi-concept cars with alternative propulsion, such as hybrid-electric, fully electric, and fuel-cell powertrains. At the 2007 Frankfurt motor show, Mercedes-Benz showed seven hybrid models, including the F700 concept car, powered by a hybrid-electric drivetrain featuring the DiesOtto engine. In 2009, Mercedes-Benz displayed three BlueZERO concepts at the North American International Auto Show. Each car features a different powertrain - battery-electric, fuel-cell electric, and gasoline-electric hybrid. In the same year, Mercedes also showed the Vision S500 PHEV concept with a all-electric range and emissions of 74 grams/km in the New European Driving Cycle.


title: Mercedes-Benz

text: Since 2002, Mercedes-Benz has developed the F-Cell fuel cell vehicle. The current version, based on the B-Class, has a 250-mile range and is available for lease, with volume production scheduled to begin in 2014. Mercedes has also announced the SLS AMG E-Cell, a fully electric version of the SLS sports car, with deliveries expected in 2013. The Mercedes-Benz S400 BlueHYBRID was launched in 2009, and is the first production automotive hybrid in the world to use a lithium-ion battery. In mid-2010, production commenced on the Vito E-Cell all-electric van.


title: Mercedes-Benz

text: In 2008, Mercedes-Benz announced that it would have a demonstration fleet of small electric cars in two to three years. Mercedes-Benz and Smart are preparing for the widespread uptake of electric vehicles (EVs) in the UK by beginning the installation of recharging points across their dealer networks. So far 20 Elektrobay recharging units, produced in the UK by Brighton-based Elektromotive, have been installed at seven locations as part of a pilot project, and further expansion of the initiative is planned later in 2010.


title: Mercedes-Benz

text: In the United States, Mercedes-Benz was assessed a record US$30.66 million fine for their decision to not meet the federal corporate average fuel economy standard in 2009. Certain Mercedes-Benz cars, including the S550 and all AMG models sold in the United States, also face an additional gas guzzler tax. However, newer AMG models fitted with the M157 engine will not be subject to the gas-guzzler tax, due to improved fuel economy, and newer models powered by the M276 and M278 engines will have better fuel economy. In 2008, Mercedes also had the worst average of all major European manufacturers, ranking 14th out of 14 manufacturers. Mercedes was also the worst manufacturer in 2007 and 2006 in terms of average levels, with 181 g and 188 g of emitted per km, respectively.


title: Mercedes-Benz

text: Mercedes-Benz paid an additional US$38 million for failing to meet its CAFE standards for model years 2008–2011.


title: Mercedes-Benz

text: In February 2018, it was announced that Mercedes cabin air filters earned the Asthma and Allergy Friendly Certification.


title: Mercedes-Benz

text: Mercedes opened its sixth battery factory in 2018. Critics deemed the marque's EQS sedan a tough competitor to Tesla, Inc. The six factories are established across three continents.


title: Mercedes-Benz

text: The brand has also launched its electric EQ brand with the EQC SUV which was set for production in the year 2019. In September 2018, Mercedes unveiled the EQC, its first fully electric car, at an event in Stockholm.


title: Mercedes-Benz

text: Mercedes unveiled the VISION AVTR at the 2020 Consumer Electronics Show in Las Vegas. The car was inspired by the 2009 science fiction film Avatar. Whilst the AVTR is a concept car, it is said to "hold clues on features bound for next-generation Mercedes-Benz cars".


title: Mercedes-Benz

text: 2022 will be the year in which Daimler has said that the company will have invested $11 billion to ensure that every Mercedes-Benz has a fully electric or hybrid version available on the market.


title: Mercedes-Benz

text: ""Our electric vehicles will be built in six plants on three continents. We address every market segment: from the smart fortwo seater to the large SUV. The battery is the key component of e-mobility. As batteries are the heart of our electric vehicles we put a great emphasis on building them in our own factories. With our global battery network, we are in an excellent position: As we are close to our vehicle plants we can ensure the optimal supply of production. In case of a short-term high demand in another part of the world our battery factories are also well prepared for export. The electric initiative of Mercedes-Benz Cars is right on track. Our global production network is ready for e-mobility. We are electrifying the future."" After Audi declared that it would cut more than 9,000 jobs by 2025, the owner of Mercedes-Benz announced that the company will shed around 10,000 jobs worldwide to focus on electric cars.


title: Mercedes-Benz

text: In January 2021, Mercedes-Benz revealed its new electric SUV, the EQA, which will have a range of 426 kilometres and will be on sale in Europe starting 4 February.


title: Mercedes-Benz

text: The two companies which were merged to form the "Mercedes-Benz" brand in 1926 had both already enjoyed success in the new sport of motor racing throughout their separate histories. A single Benz competed in the world's first motor race, the 1894 Paris–Rouen, where Émile Roger finished 14th in 10 hours 1 minute. Throughout its long history, the company has been involved in a range of motorsport activities, including sports car racing and rallying. On several occasions, Mercedes-Benz has withdrawn completely from motorsport for a significant period, notably in the late 1930s, and after the 1955 Le Mans disaster, where a Mercedes-Benz 300 SLR rammed another car (an Austin-Healey), took off into the stands, and killed more than 80 spectators. Stirling Moss and co-driver Denis Jenkinson won the 1955 Mille Miglia road race in Italy during a record-breaking drive with an average speed of almost 98 mph in a Mercedes-Benz 300 SLR.


title: Mercedes-Benz

text: Although there was some activity in the intervening years, it was not until 1987 that Mercedes-Benz returned to front line competition, returning to Le Mans, Deutsche Tourenwagen Meisterschaft (DTM), and Formula One with Sauber. The 1990s saw Mercedes-Benz partner with British engine builder Ilmor (now Mercedes-Benz High Performance Engines), and campaign IndyCars under the USAC/CART rules, eventually winning the 1994 Indianapolis 500 and 1994 CART IndyCar World Series Championship with Al Unser Jr. at the wheel. The 1990s also saw the return of Mercedes-Benz to GT racing with the Mercedes-Benz CLK GTR, which took the company to two titles in FIA's GT1 class.


title: Mercedes-Benz

text: Mercedes-Benz is currently active in three motorsport categories, Formula E, Formula One and GT racing.


title: Mercedes-Benz

text: Mercedes-Benz took part in the world championship in 1954 and 1955, but despite being successful with two championship titles for Juan-Manuel Fangio, the company left the sport after just two seasons.


title: Mercedes-Benz

text: Mercedes-Benz returned as an engine manufacturer in 1994, with the engines being designed and manufactured by Ilmor in Brixworth. It initially partnered with Sauber, before switching to McLaren in 1995. Although the Mercedes engines were not successful at first, they later won drivers' championships for Mika Häkkinen in 1998 and 1999, and for Lewis Hamilton in 2008, as well as a constructors' championship in 1998. Mercedes part-owned McLaren, and the collaboration had been extended into the production of road-going cars such as the Mercedes-Benz SLR McLaren.


title: Mercedes-Benz

text: In 2007, McLaren-Mercedes was fined a record US$100 million for stealing confidential Ferrari technical data.


title: Mercedes-Benz

text: In 2009, Ross Brawn's newly conceived Formula One team, Brawn GP used Mercedes engines to win the titles. At the end of the season, Mercedes-Benz sold its 40% stake in McLaren to the McLaren Group and bought 70% of the Brawn GP team jointly with an Abu Dhabi-based investment consortium. Brawn GP was renamed Mercedes GP for the 2010 season and became the main team for Mercedes-Benz. The company continued providing engines to other teams under customer relationships.


title: Mercedes-Benz

text: After major rule changes in , Mercedes clinched the drivers' and constructors' titles with drivers Lewis Hamilton and Nico Rosberg, after dominating much of the season. Mercedes repeated the feat in , winning 16 out of 19 races, and again in , winning 19 of the 21 races. In the following years, Mercedes continued their success by winning the drivers' championships from 2017 to 2020 and the constructors' championships from 2017 to 2021, becoming the first team to win seven consecutive "double-championships". In these years, Hamilton was the champion in 2014, 2015, 2017, 2018, 2019 and 2020, while Rosberg won in 2016. Their unbeaten streak was broken in 2021, when Max Verstappen of Red Bull-Honda won the drivers' championship.


title: Mercedes-Benz

text: Prior to pre-season testing of the 2019–20 Formula E Championship, it was announced that Mercedes, through its EQ branch, would join the championship with drivers Stoffel Vandoorne and 2019 FIA Formula 2 champion Nyck De Vries. The team named their Spark Gen2 challenger the Mercedes EQ Silver Arrow 01.


title: Mercedes-Benz

text: In June 1909, Daimler Motoren Gesellschaft (DMG) registered both a three-pointed and a four-pointed star as trademarks, but only the three-pointed star was used. To DMG, the star symbolized Gottlieb Daimler's aims for universal motorization: on land, water and in the air.


title: Mercedes-Benz

text: Numerous technological innovations have been introduced on Mercedes-Benz automobiles throughout the many years of their production, including:


title: Mercedes-Benz

text: In the 1980s Mercedes built the world's first robot car, together with the team of Professor Ernst Dickmanns at Bundeswehr University Munich. Encouraged in part by Dickmanns' success, in 1987 the European Union's EUREKA programme initiated the Prometheus Project on autonomous vehicles, funded to the tune of nearly €800 million. In 1995 Dickmanns' re-engineered autonomous S-Class Mercedes took a long trip from Munich in Bavaria to Copenhagen in Denmark, and back. On highways, the robot achieved speeds exceeding (permissible in some areas of the German Autobahn).


title: Mercedes-Benz

text: In October 2015, the company introduced the Vision Tokyo, a five-seat self-driving electric van powered by a hybrid hydrogen fuel-cell system. The super-sleek van is touted as "a chill-out zone in the midst of megacity traffic mayhem."


title: Mercedes-Benz

text: Several companies have become car tuners (or modifiers) of Mercedes Benz, in order to increase performance and/or luxury to a given model.


title: Mercedes-Benz

text: AMG is Mercedes-Benz's in-house performance-tuning division, specialising in high-performance versions of most Mercedes-Benz cars. AMG engines are all hand-built, and each completed engine receives a tag with the signature of the engineer who built it. AMG has been wholly owned by Mercedes-Benz since 1999. The 2009 SLS AMG, a revival of the 300SL Gullwing, is the first car to be entirely developed by AMG.


title: Mercedes-Benz

text: In football (soccer), Mercedes-Benz sponsored the Germany national team until 2018. Mercedes-Benz sponsors Bundesliga club VfB Stuttgart and provides the naming rights for their stadium, the Mercedes-Benz Arena. The company formerly held a ten-year naming rights contract to the Caesars Superdome, an American football stadium in New Orleans, from 2011 to 2021. On 24 August 2015, Mercedes-Benz was announced as the naming rights sponsor for the Atlanta Falcons' new home, Mercedes-Benz Stadium, (Mercedes-Benz's US headquarters are in Greater Atlanta) which opened in August 2017.


title: Mercedes-Benz

text: Mercedes-Benz worked with English magician Steven Frayne, also known as Dynamo, to create a video called Dynamo vs Coulthard. Formula One driver David Coulthard drove Dynamo around a track at race-speed in a Mercedes-Benz SL63 AMG, before Dynamo successfully drove around the same track in the same car whilst blindfolded and surrounded by pyrotechnics. The stunt was part of the finale for Series 3 of , screened on UK television channel Watch.


title: Mercedes-Benz

text: To promote the release of "Mario Kart 8", Nintendo and Mercedes-Benz partnered up to have 3 of their cars in their history, the 2014 Mercedes-Benz GLA, the 1957 SL 300 Roadster, and the 1934 W25 Silver Arrow as karts in the game. As part of the partnership, Mario, Luigi, and Peach appeared in Japanese commercials to promote the 2014 Mercedes-Benz GLA.


title: Mercedes-Benz

text: The Mercedes-Benz was chosen by Mitsubishi Electric as an official sponsor of AFF Mitsubishi Electric Cup 2022, along with Maspion Holdings, Wuling, Gree, Yanmar and Tiger Brokers.


title: Good Will Hunting

text: Good Will Hunting is a 1997 American psychological drama film directed by Gus Van Sant, and written by Ben Affleck and Matt Damon. It stars Robin Williams, Damon, Affleck, Stellan Skarsgård and Minnie Driver.


title: Good Will Hunting

text: The film received positive reviews from critics and grossed over $225 million during its theatrical run against a $10 million budget. At the 70th Academy Awards, it received nominations in nine categories, including Best Picture and Best Director, and won in two: Best Supporting Actor for Williams and Best Original Screenplay for Affleck and Damon. In 2014, it was ranked at number 53 in "The Hollywood Reporter"s "100 Favorite Films" list.


title: Good Will Hunting

text: Twenty-year-old Will Hunting of South Boston is a natural genius who is self-taught and been recently paroled from jail. He works as a janitor at MIT and spends his free time drinking with his step-brother Chuckie and his friends Billy and Morgan. When Professor Gerald Lambeau posts a difficult combinatorial mathematics problem on a blackboard as a challenge for his graduate students, Will solves the problem anonymously, stunning both the students and Lambeau. As a challenge to the unknown genius, Lambeau posts an even more difficult problem; he later catches Will writing the solution on the blackboard late at night, but initially thinks Will is vandalizing it and chases him off.


title: Good Will Hunting

text: At a bar, Will meets Skylar, a British woman about to graduate from Harvard College, who plans on attending medical school at Stanford.


title: Good Will Hunting

text: The next day, Will and his friends fight a gang that contains a member who used to bully Will as a child. Will is arrested after he attacks a responding police officer. Lambeau sits in on his court appearance and watches Will defend himself. He arranges for him to avoid jail time if he agrees to study mathematics under Lambeau's supervision and participate in psychotherapy sessions. Will tentatively agrees but treats his therapists with mockery. In desperation, Lambeau calls on Dr. Sean Maguire, his college roommate, who now teaches psychology at Bunker Hill Community College. Unlike other therapists, Sean actually challenges Will's defense mechanisms. During the first session, Will insults Sean's deceased wife, and Sean threatens him—but after a few unproductive sessions, Will finally begins to open up.


title: Good Will Hunting

text: Will is particularly struck by Sean's story of how he met his wife, who later died of cancer, by giving up his ticket to the historic game six of the 1975 World Series, after falling in love at first sight. Sean's explanation for surrendering his ticket was to "see about a girl," and he does not regret his decision. This encourages Will to build a relationship with Skylar, though he lies to her about his past and is reluctant to introduce her to his friends or show her his rundown neighborhood. Will also challenges Sean to take an objective look at his own life, since Sean cannot move on from his wife's death.


title: Good Will Hunting

text: Lambeau sets up a number of job interviews for Will, but Will scorns them by sending Chuckie as his "chief negotiator", and by turning down a position at the NSA with a scathing critique of the agency's moral position. Skylar asks Will to move to California with her, but he refuses and tells her he is an orphan, and that his foster father physically abused him. Will breaks up with Skylar and later storms out on Lambeau, dismissing the mathematical research he has been doing. Sean points out that Will is so adept at anticipating future failure in his interpersonal relationships that he deliberately sabotages them in order to avoid emotional pain. Chuckie likewise challenges Will over his resistance to taking any of the positions he interviews for, telling Will he owes it to his friends to make the most of opportunities they will never have, even if it means leaving one day. He then tells Will that the best part of his day is a brief moment when he waits on his doorstep thinking Will has moved on to something greater.


title: Good Will Hunting

text: Will walks in on a heated argument between Sean and Lambeau over his potential. Sean and Will share and find out that they were both victims of child abuse. Sean helps Will to see that he is a victim of his own inner demons and to accept that it is not his fault, causing him to break down in tears in Sean’s arms. Will accepts one of the job offers arranged by Lambeau. Having helped Will overcome his problems, Sean reconciles with Lambeau, deciding to take a sabbatical. Will's friends present him with a Chevrolet Nova for his 21st birthday so he can commute to work. Later, Chuckie goes to Will's house to pick him up, only to find that he is not there, much to his happiness. Will sends Sean a letter telling him to tell Lambeau that he had to go "see about a girl," revealing he passed on the job offer and instead is heading to California to reunite with Skylar.


title: Good Will Hunting

text: Matt Damon started writing the film as a final assignment for a playwriting class he was taking at Harvard University. Instead of writing a one-act play, Damon submitted a 40-page script. He wrote his then-girlfriend, medical student Skylar Satenstein (credited in the closing credits of the film), into his script. Damon said the only scene from that script that survived – "it survived verbatim" – was when Will Hunting (Damon) meets his therapist, Sean Maguire (Robin Williams), for the first time. Damon asked Ben Affleck to develop the screenplay together. They completed the script in 1994. At first, it was written as a thriller about a young man in the rough-and-tumble streets of South Boston who possesses a superior intelligence and is targeted by the government with heavy-handed recruitment.


title: Good Will Hunting

text: Castle Rock Entertainment bought the script for $675,000 against $775,000, meaning that Damon and Affleck would stand to earn an additional $100,000 if the film was produced and they retained sole writing credit. Castle Rock president Rob Reiner urged them to drop the thriller aspect of the story and to focus on the relationship between Will and his therapist. Terrence Malick told Affleck and Damon over dinner that the film ought to end with Will's decision to follow his girlfriend Skylar to California, not them leaving together.


title: Good Will Hunting

text: At Reiner's request, screenwriter William Goldman read the script. Goldman consistently denied the persistent rumor that he wrote "Good Will Hunting" or acted as a script doctor. In his book "Which Lie Did I Tell?" Goldman jokingly writes, "I did not just doctor it. I wrote the whole thing from scratch," before dismissing the rumor as false and saying his only advice was agreeing with Reiner's suggestion.


title: Good Will Hunting

text: Affleck and Damon proposed to act in the lead roles, but many studio executives said they wanted Brad Pitt and Leonardo DiCaprio. Meanwhile, Kevin Smith was working with Affleck on "Mallrats" and with both Damon and Affleck on "Chasing Amy." Castle Rock put the script in turnaround, and gave Damon and Affleck 30 days to find another buyer for the script who would reimburse Castle Rock the money paid, otherwise the script reverted to the studio, and Damon and Affleck would be out. All the studios that were involved in the original bidding war for the screenplay now turned the pair down, taking meetings with Affleck and Damon only to tell them this to their face. As a last resort, Affleck passed the script to his "Chasing Amy" director Kevin Smith, who read it and promised to walk the script directly into Harvey Weinstein's office at Miramax. Weinstein read the script, loved it, and paid Castle Rock their due, while also agreeing to let Damon and Affleck star in the film. Weinstein asked that a few scenes be removed, including an out-of-place, mid-script oral sex scene that Damon and Affleck added to trick executives who were not looking closely.


title: Good Will Hunting

text: After buying the rights from Castle Rock, Miramax put the film into production. Several well-known filmmakers were originally considered to direct, including Mel Gibson and Michael Mann. Originally, Affleck asked Kevin Smith whether he was interested in directing. He declined, saying they needed a "good director," that he only directed projects he wrote, and that he was not much of a visual director, but he still served as one of the film's executive producers. Damon and Affleck later chose Gus Van Sant for the job, whose work on previous films like "Drugstore Cowboy" (1989) had left a favorable impression on the fledgling screenwriters. Miramax was persuaded and hired Van Sant to direct the film.


title: Good Will Hunting

text: Filming took place between April and June 1997. Although the story is set in Boston, and many of the scenes were shot on location in the Greater Boston area, many of the interior shots were filmed at locations in Toronto, with the University of Toronto standing in for MIT and Harvard University. The classroom scenes were filmed at McLennan Physical Laboratories (of the University of Toronto) and Central Technical School. Harvard normally disallows filming on its property, but permitted limited filming by the project after intervention by Harvard alumnus John Lithgow. Likewise, only the exterior shots of Bunker Hill Community College were filmed in Boston; however, Sean's office was built in Toronto as an exact replica of one at the college.


title: Good Will Hunting

text: The interior bar scenes set in "Southie" were shot on location at Woody's L Street Tavern. Meanwhile, the homes of Will (190 West 6th Street) and Sean (259 E Street), while some distance apart in the movie, actually back up to each other on Bowen Street, the narrow street Chuckie drives down to walk up to Will's back door.


title: Good Will Hunting

text: The Bow and Arrow Pub, which was located at the corner of Bow Street and Massachusetts Avenue in Cambridge, doubled as the exterior of the Harvard bar in which Will met Skylar for the first time. The Baskin-Robbins/Dunkin' Donuts featured in the "How do you like "them" apples?" scene was next door to the pub at the time of the film's release. The Harvard Bar interior scenes were filmed at the Upfront Bar and Grill on Front St. E. in Toronto.


title: Good Will Hunting

text: The Tasty, at the corner of JFK and Brattle Streets, was the scene of Will and Skylar's first kiss. The Au Bon Pain, where Will and Skylar discuss the former's photographic memory, was at the corner of Dunster Street and Mass Ave.


title: Good Will Hunting

text: The Boston Public Garden bench on which Will and Sean sat for a scene in the film became a temporary shrine after Williams's death in 2014.