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In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two.
This theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th. It is among the most famous theorems in the history of mathematics and prior to its 1995 proof was in the Guinness Book of World Records for "most difficult math problems".
1 Fermat's conjecture (History)
2 Mathematical context
2.1 Pythagorean triples
2.2 Diophantine equations
3 Fermat's conjecture
4 Proofs for specific exponents
5 Sophie Germain
6 Ernst Kummer and the theory of ideals
7 Mordell conjecture
8 Rational exponents
9 Computational studies
10 Connection with elliptic curves
11 Wiles' general proof
12 Did Fermat possess a general proof?
13 Monetary prizes
14 In popular culture
15 See also
19 Further reading
20 External links
 Fermat's conjecture (History)
Fermat left no proof of the conjecture for all n, but he did prove the special case n = 4. This reduced the problem to proving the theorem for exponents n that are prime numbers. Over the next two centuries (1637–1839), the conjecture was proven for only the primes 3, 5, and 7, although Sophie Germain proved a special case for all primes less than 100. In the mid-19th century, Ernst Kummer proved the theorem for regular primes. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to prove the conjecture for all odd primes up to four million.
The final proof of the conjecture for all n came in the late 20th century. In 1984, Gerhard Frey suggested the approach of proving the conjecture through a proof of the modularity theorem for elliptic curves. Building on work of Ken Ribet, Andrew Wiles succeeded in proving enough of the modularity theorem to prove Fermat's Last Theorem, with the assistance of Richard Taylor. Wiles's achievement was reported widely in the popular press, and has been popularized in books and television programs.
 Mathematical context
 Pythagorean triples
Main article: Pythagorean triple
Pythagorean triples are a set of three integers (a, b, c) that satisfy a special case of Fermat's equation (n = 2)
a^2 + b^2 = c^2.\
Examples of Pythagorean triples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples, and methods for generating such triples have been studied in many cultures, beginning with the Babylonians and later ancient Greek, Chinese and Indian mathematicians. The traditional interest in Pythagorean triples connects with the Pythagorean theorem; in its converse form, it states that a triangle with sides of lengths a, b and c has a right angle between the a and b legs when the numbers are a Pythagorean triple. Right angles have various practical applications, such as surveying, carpentry, masonry and construction. Fermat's Last Theorem is an extension of this problem to higher powers, stating that no solution exists when the exponent 2 is replaced by any larger integer.
 Diophantine equations
Main article: Diophantine equation
Fermat's equation xn + yn = zn is an example of a Diophantine equation. A Diophantine equation is a polynomial equation in which the solutions must be integers. Their name derives from the 3rd-century Alexandrian mathematician, Diophantus, who developed methods for their solution. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively:
A = x + y\
B = x^2 + y^2\
Diophantus' major work is the Arithmetica, of which only a portion has survived. Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the Arithmetica, which was translated into Latin and published in 1621 by Claude Bachet.
Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC). Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC). Many Diophantine equations have a form similar to the equation of Fermat's Last Theorem from the point of view of algebra, in that they have no cross terms mixing two letters, without sharing its particular properties. For example, it is known that there are infinitely many positive integers x, y, and z such that xn + yn = zm where n and m are relatively prime natural numbers.[note 1]
 Fermat's conjecture
Problem II.8 in the 1621 edition of the Arithmetica of Diophantus. On the right is the famous margin which was too small to contain Fermat's alleged proof of his "last theorem".
Problem II.8 of the Arithmetica asks how to split a given square number into two other squares; in other words, for a given rational number k, find rational numbers u and v such that k2 = u2 + v2. Diophantus shows how to solve this sum-of-squares problem for k = 4 (the solutions being u = 16/5 and v = 12/5).
Around 1637, Fermat wrote his Last Theorem in the margin of his copy of the Arithmetica next to Diophantus' sum-of-squares problem:
Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.
Although Fermat's general proof is unknown, his proof of one case (n = 4) by infinite descent has survived. Fermat posed the cases of n = 4 and of n = 3 as challenges to his mathematical correspondents, such as Marin Mersenne, Blaise Pascal, and John Wallis. However, in the last thirty years of his life, Fermat never again wrote of his "truly marvellous proof" of the general case.
After Fermat's death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father's comments. The margin note became known as Fermat's Last Theorem, as it was the last of Fermat's asserted theorems to remain unproven.
 Proofs for specific exponents
Main article: Proof of Fermat's Last Theorem for specific exponents
Only one mathematical proof by Fermat has survived, in which Fermat uses the technique of infinite descent to show that the area of a right triangle with integer sides can never equal the square of an integer. His proof is equivalent to demonstrating that the equation
x4 - y4 = z2
has no primitive solutions in integers (no pairwise coprime solutions). In turn, this proves Fermat's Last Theorem for the case n=4, since the equation a4 + b4 = c4 can be written as c4 - b4 = (a2)2. For a version of Fermat's proof by infinite descent, see Infinite descent#Non-solvability of r2 + s4 = t4. For various proofs by infinite descent, see Grant and Perella (1999), Barbara (2007), and Dolan (2011).
Alternative proofs of the case n = 4 were developed later by Frénicle de Bessy (1676), Leonhard Euler (1738), Kausler (1802), Peter Barlow (1811), Adrien-Marie Legendre (1830), Schopis (1825), Terquem (1846), Joseph Bertrand (1851), Victor Lebesgue (1853, 1859, 1862), Theophile Pepin (1883), Tafelmacher (1893), David Hilbert (1897), Bendz (1901), Gambioli (1901), Leopold Kronecker (1901), Bang (1905), Sommer (1907), Bottari (1908), Karel Rychlík (1910), Nutzhorn (1912), Robert Carmichael (1913), Hancock (1931), and Vranceanu (1966).
After Fermat proved the special case n = 4, the general proof for all n required only that the theorem be established for all odd prime exponents. In other words, it was necessary to prove only that the equation an + bn = cn has no integer solutions (a, b, c) when n is an odd prime number. This follows because a solution (a, b, c) for a given n is equivalent to a solution for all the factors of n. For illustration, let n be factored into d and e, n = de. The general equation
an + bn = cn
implies that (ad, bd, cd) is a solution for the exponent e
(ad)e + (bd)e = (cd)e
Thus, to prove that Fermat's equation has no solutions for n > 2, it suffices to prove that it has no solutions for at least one prime factor of every n. All integers n > 2 contain a factor of 4, or an odd prime number, or both. Therefore, Fermat's Last Theorem can be proven for all n if it can be proven for n = 4 and for all odd primes (the only even prime number is the number 2) p.
In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proven for three odd prime exponents p = 3, 5 and 7. The case p = 3 was first stated by Abu-Mahmud Khojandi (10th century), but his attempted proof of the theorem was incorrect. In 1770, Leonhard Euler gave a proof of p = 3, but his proof by infinite descent contained a major gap. However, since Euler himself had proven the lemma necessary to complete the proof in other work, he is generally credited with the first proof. Independent proofs were published by Kausler (1802), Legendre (1823, 1830), Calzolari (1855), Gabriel Lamé (1865), Peter Guthrie Tait (1872), Günther (1878), Gambioli (1901), Krey (1909), Rychlík (1910), Stockhaus (1910), Carmichael (1915), Johannes van der Corput (1915), Axel Thue (1917), and Duarte (1944). The case p = 5 was proven independently by Legendre and Peter Dirichlet around 1825. Alternative proofs were developed by Carl Friedrich Gauss (1875, posthumous), Lebesgue (1843), Lamé (1847), Gambioli (1901), Werebrusow (1905), Rychlík (1910), van der Corput (1915), and Guy Terjanian (1987). The case p = 7 was proven by Lamé in 1839. His rather complicated proof was simplified in 1840 by Lebesgue, and still simpler proofs were published by Angelo Genocchi in 1864, 1874 and 1876. Alternative proofs were developed by Théophile Pépin (1876) and Edmond Maillet (1897).
Fermat's Last Theorem has also been proven for the exponents n = 6, 10, and 14. Proofs for n = 6 have been published by Kausler, Thue, Tafelmacher, Lind, Kapferer, Swift, and Breusch. Similarly, Dirichlet and Terjanian each proved the case n = 14, while Kapferer and Breusch each proved the case n = 10. Strictly speaking, these proofs are unnecessary, since these cases follow from the proofs for n = 3, 5, and 7, respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. Dirichlet's proof for n = 14 was published in 1832, before Lamé's 1839 proof for n = 7.
Many proofs for specific exponents use Fermat's technique of infinite descent, which Fermat used to prove the case n = 4, but many do not. However, the details and auxiliary arguments are often ad hoc and tied to the individual exponent under consideration. Since they became ever more complicated as p increased, it seemed unlikely that the general case of Fermat's Last Theorem could be proven by building upon the proofs for individual exponents. Although some general results on Fermat's Last Theorem were published in the early 19th century by Niels Henrik Abel and Peter Barlow, the first significant work on the general theorem was done by Sophie Germain.
 Sophie Germain
Main article: Sophie Germain
In the early 19th century, Sophie Germain developed several novel approaches to prove Fermat's last theorem for all exponents. First, she defined a set of auxiliary primes ? constructed from the prime exponent p by the equation ? = 2hp+1, where h is any integer not divisible by three. She showed that if no integers raised to the pth power were adjacent modulo ? (the non-consecutivity condition), then ? must divide the product xyz. Her goal was to use mathematical induction to prove that, for any given p, infinitely many auxiliary primes ? satisfied the non-consecutivity condition and thus divided xyz; since the product xyz can have at most a finite number of prime factors, such a proof would have established Fermat's Last Theorem. Although she developed many techniques for establishing the non-consecutivity condition, she did not succeed in her strategic goal. She also worked to set lower limits on the size of solutions to Fermat's equation for a given exponent p, a modified version of which was published by Adrien-Marie Legendre. As a byproduct of this latter work, she proved Sophie Germain's theorem, which verified the first case of Fermat's Last Theorem for every odd prime exponent less than 100. Germain tried unsuccessfully to prove the first case of Fermat's Last Theorem for all even exponents, specifically for n = 2p, which was proven by Guy Terjanian in 1977. In 1985, Leonard Adleman, Roger Heath-Brown and Étienne Fouvry proved that the first case of Fermat's Last Theorem holds for infinitely many odd primes p.
 Ernst Kummer and the theory of ideals
In 1847, Gabriel Lamé outlined a proof of Fermat's Last Theorem based on factoring the equation xp + yp = zp in complex numbers, specifically the cyclotomic field based on the roots of the number 1. His proof failed, however, because it assumed incorrectly that such complex numbers can be factored uniquely into primes, similar to integers. This gap was pointed out immediately by Joseph Liouville, who later read a paper that demonstrated this failure of unique factorisation, written by Ernst Kummer.
Kummer set himself the task of determining whether the cyclotomic field could be generalized to include new prime numbers such that unique factorisation was restored. He succeeded in that task by developing the ideal numbers. Using the general approach outlined by Lamé, Kummer proved both cases of Fermat's Last Theorem for all regular prime numbers. However, he could not prove the theorem for the exceptional primes (irregular primes) which conjecturally occur approximately 39% of the time; the only irregular primes below 100 are 37, 59 and 67.
 Mordell conjecture
In the 1920s, Louis Mordell posed a conjecture that implied that Fermat's equation has at most a finite number of nontrivial primitive integer solutions if the exponent n is greater than two. This conjecture was proven in 1983 by Gerd Faltings, and is now known as Faltings' theorem.
 Rational exponents
All solutions of the Diophantine equation an / m + bn / m = cn / m when n=1 were computed by Lenstra in 1992. In 2004, for n>2, Bennett, Glass, and Szekely proved that if gcd(n,m)=1, then there are integer solutions if and only if 6 divides m, and a1 / m, b1 / m, and c1 / m are different complex 6th roots of the same real number.
 Computational studies
In the latter half of the 20th century, computational methods were used to extend Kummer's approach to the irregular primes. In 1954, Harry Vandiver used a SWAC computer to prove Fermat's Last Theorem for all primes up to 2521. By 1978, Samuel Wagstaff had extended this to all primes less than 125,000. By 1993, Fermat's Last Theorem had been proven for all primes less than four million.
 Connection with elliptic curves
The ultimately successful strategy for proving Fermat's Last Theorem was by proving the modularity theorem. The strategy was first described by Gerhard Frey in 1984. Frey noted that if Fermat's equation had a solution (a, b, c) for exponent p > 2, the corresponding elliptic curve[note 2]
y2 = x (x - ap)(x + bp)
would have such unusual properties that the curve would likely violate the modularity theorem. This theorem, first conjectured in the mid-1950s and gradually refined through the 1960s, states that every elliptic curve is modular, meaning that it can be associated with a unique modular form.
Following this strategy, the proof of Fermat's Last Theorem required two steps. First, it was necessary to show that Frey's intuition was correct, that the above elliptic curve is always non-modular. Frey did not succeed in proving this rigorously; the missing piece was identified by Jean-Pierre Serre. This missing piece, the so-called "epsilon conjecture", was proven by Ken Ribet in 1986. Second, it was necessary to prove a special case of the modularity theorem. This special case (for semistable elliptic curves) was proven by Andrew Wiles in 1995.
Thus, the epsilon conjecture showed that any solution to Fermat's equation could be used to generate a non-modular semistable elliptic curve, whereas Wiles' proof showed that all such elliptic curves must be modular. This contradiction implies that there can be no solutions to Fermat's equation, thus proving Fermat's Last Theorem.
 Wiles' general proof
British mathematician Andrew Wiles
Main article: Wiles' proof of Fermat's Last Theorem
Ribet's proof of the epsilon conjecture in 1986 accomplished the first half of Frey's strategy for proving Fermat's Last Theorem. Upon hearing of Ribet's proof, Andrew Wiles decided to commit himself to accomplishing the second half: proving a special case of the modularity theorem (then known as the Taniyama–Shimura conjecture) for semistable elliptic curves. Wiles worked on that task for six years in almost complete secrecy. He based his initial approach on his area of expertise, Horizontal Iwasawa theory, but by the summer of 1991, this approach seemed inadequate to the task. In response, he exploited an Euler system recently developed by Victor Kolyvagin and Matthias Flach. Since Wiles was unfamiliar with such methods, he asked his Princeton colleague, Nick Katz, to check his reasoning over the spring semester of 1993.
By mid-1993, Wiles was sufficiently confident of his results that he presented them in three lectures delivered on June 21–23, 1993 at the Isaac Newton Institute for Mathematical Sciences. Specifically, Wiles presented his proof of the Taniyama–Shimura conjecture for semistable elliptic curves; together with Ribet's proof of the epsilon conjecture, this implied Fermat's Last Theorem. However, it soon became apparent that Wiles' initial proof was incorrect. A critical portion of the proof contained an error in a bound on the order of a particular group. The error was caught by several mathematicians refereeing Wiles' manuscript, including Katz, who alerted Wiles on 23 August 1993.
Wiles and his former student Richard Taylor spent almost a year trying to repair the proof, without success. On 19 September 1994, Wiles had a flash of insight that the proof could be saved by returning to his original Horizontal Iwasawa theory approach, which he had abandoned in favour of the Kolyvagin–Flach approach. On 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem" and "Ring theoretic properties of certain Hecke algebras", the second of which was co-authored with Taylor. The two papers were vetted and published as the entirety of the May 1995 issue of the Annals of Mathematics. These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.
 Did Fermat possess a general proof?
The mathematical techniques used in Fermat's "marvelous" proof are unknown. Only one detailed proof of Fermat has survived, the above proof that no three coprime integers (x, y, z) satisfy the equation x4 - y4 = z4.
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Google began in January 1996 as a research project by Larry Page and Sergey Brin when they were both PhD students at Stanford University in California.
While conventional search engines ranked results by counting how many times the search terms appeared on the page, the two theorized about a better system that analyzed the relationships between websites. They called this new technology PageRank, where a website's relevance was determined by the number of pages, and the importance of those pages, that linked back to the original site.
A small search engine called "RankDex" from IDD Information Services designed by Robin Li was, since 1996, already exploring a similar strategy for site-scoring and page ranking. The technology in RankDex would be patented and used later when Li founded Baidu in China.
Page and Brin originally nicknamed their new search engine "BackRub", because the system checked backlinks to estimate the importance of a site.
Eventually, they changed the name to Google, originating from a misspelling of the word "googol", the number one followed by one hundred zeros, which was picked to signify that the search engine wants to provide large quantities of information for people. Originally, Google ran under the Stanford University website, with the domain google.stanford.edu.
The domain name for Google was registered on September 15, 1997, and the company was incorporated on September 4, 1998. It was based in a friend's (Susan Wojcicki) garage in Menlo Park, California. Craig Silverstein, a fellow PhD student at Stanford, was hired as the first employee.
In May 2011, unique visitors of Google surpassed 1 billion mark for the first time, an 8.4 percent increase from a year ago with 931 million unique visitors.
Financing and initial public offering
Google's first servers, showing lots of exposed wiring and circuit boards
The first iteration of Google production servers was built with inexpensive hardware
The first funding for Google was an August 1998 contribution of US$100,000 from Andy Bechtolsheim, co-founder of Sun Microsystems, given before Google was even incorporated. Early in 1999, while still graduate students, Brin and Page decided that the search engine they had developed was taking up too much of their time from academic pursuits. They went to Excite CEO George Bell and offered to sell it to him for $1 million. He rejected the offer, and later criticized Vinod Khosla, one of Excite's venture capitalists, after he had negotiated Brin and Page down to $750,000. On June 7, 1999, a $25 million round of funding was announced, with major investors including the venture capital firms Kleiner Perkins Caufield & Byers and Sequoia Capital.
Google's initial public offering (IPO) took place five years later on August 19, 2004. The company offered 19,605,052 shares at a price of $85 per share. Shares were sold in a unique online auction format using a system built by Morgan Stanley and Credit Suisse, underwriters for the deal. The sale of $1.67 billion gave Google a market capitalization of more than $23 billion. The vast majority of the 271 million shares remained under the control of Google, and many Google employees became instant paper millionaires. Yahoo!, a competitor of Google, also benefited because it owned 8.4 million shares of Google before the IPO took place.
Some people speculated that Google's IPO would inevitably lead to changes in company culture. Reasons ranged from shareholder pressure for employee benefit reductions to the fact that many company executives would become instant paper millionaires. As a reply to this concern, co-founders Sergey Brin and Larry Page promised in a report to potential investors that the IPO would not change the company's culture. In 2005, however, articles in The New York Times and other sources began suggesting that Google had lost its anti-corporate, no evil philosophy. In an effort to maintain the company's unique culture, Google designated a Chief Culture Officer, who also serves as the Director of Human Resources. The purpose of the Chief Culture Officer is to develop and maintain the culture and work on ways to keep true to the core values that the company was founded on: a flat organization with a collaborative environment. Google has also faced allegations of sexism and ageism from former employees.
The stock's performance after the IPO went well, with shares hitting $700 for the first time on October 31, 2007, primarily because of strong sales and earnings in the online advertising market. The surge in stock price was fueled mainly by individual investors, as opposed to large institutional investors and mutual funds. The company is now listed on the NASDAQ stock exchange under the ticker symbol GOOG and under the Frankfurt Stock Exchange under the ticker symbol GGQ1.
In March 1999, the company moved its offices to Palo Alto, California, home to several other noted Silicon Valley technology startups. The next year, against Page and Brin's initial opposition toward an advertising-funded search engine, Google began selling advertisements associated with search keywords. In order to maintain an uncluttered page design and increase speed, advertisements were solely text-based. Keywords were sold based on a combination of price bids and click-throughs, with bidding starting at five cents per click. This model of selling keyword advertising was first pioneered by Goto.com, an Idealab spin-off created by Bill Gross. When the company changed names to Overture Services, it sued Google over alleged infringements of the company's pay-per-click and bidding patents. Overture Services would later be bought by Yahoo! and renamed Yahoo! Search Marketing. The case was then settled out of court, with Google agreeing to issue shares of common stock to Yahoo! in exchange for a perpetual license.
During this time, Google was granted a patent describing its PageRank mechanism. The patent was officially assigned to Stanford University and lists Lawrence Page as the inventor. In 2003, after outgrowing two other locations, the company leased its current office complex from Silicon Graphics at 1600 Amphitheatre Parkway in Mountain View, California. The complex has since come to be known as the Googleplex, a play on the word googolplex, the number one followed by a googol zeroes. The Googleplex interiors were designed by Clive Wilkinson Architects. Three years later, Google would buy the property from SGI for $319 million. By that time, the name "Google" had found its way into everyday language, causing the verb "google" to be added to the Merriam Webster Collegiate Dictionary and the Oxford English Dictionary, denoted as "to use the Google search engine to obtain information on the Internet."
Acquisitions and partnerships
See also: List of acquisitions by Google
Since 2001, Google has acquired many companies, mainly focusing on small venture capital companies. In 2004, Google acquired Keyhole, Inc. The start-up company developed a product called Earth Viewer that gave a three-dimensional view of the Earth. Google renamed the service to Google Earth in 2005. Two years later, Google bought the online video site YouTube for $1.65 billion in stock. On April 13, 2007, Google reached an agreement to acquire DoubleClick for $3.1 billion, giving Google valuable relationships that DoubleClick had with Web publishers and advertising agencies. Later that same year, Google purchased GrandCentral for $50 million. The site would later be changed over to Google Voice. On August 5, 2009, Google bought out its first public company, purchasing video software maker On2 Technologies for $106.5 million. Google also acquired Aardvark, a social network search engine, for $50 million, and commented on its internal blog, "we're looking forward to collaborating to see where we can take it". In April 2010, Google announced it had acquired a hardware startup, Agnilux.
In addition to the many companies Google has purchased, the company has partnered with other organizations for everything from research to advertising. In 2005, Google partnered with NASA Ames Research Center to build 1,000,000 square feet (93,000 m2) of offices. The offices would be used for research projects involving large-scale data management, nanotechnology, distributed computing, and the entrepreneurial space industry. Google entered into a partnership with Sun Microsystems in October 2005 to help share and distribute each other's technologies. The company also partnered with AOL of Time Warner, to enhance each other's video search services. Google's 2005 partnerships also included financing the new .mobi top-level domain for mobile devices, along with other companies including Microsoft, Nokia, and Ericsson. Google would later launch "Adsense for Mobile", taking advantage of the emerging mobile advertising market. Increasing its advertising reach even further, Google and Fox Interactive Media of News Corporation entered into a $900 million agreement to provide search and advertising on popular social networking site MySpace.
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