Fermat's Last Theorem
In number theory Fermat's Last Theorem states that no three positive integers a, b, c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have an infinite number of solutions; the proposition was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica. However, there were first doubts about it since the publication was done by his son without his consent, after Fermat's death. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles, formally published in 1995, it proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century, it is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.
The Pythagorean equation, x2 + y2 = z2, has an infinite number of positive integer solutions for x, y, z. Around 1637, Fermat wrote in the margin of a book that the more general equation an + bn = cn had no solutions in positive integers if n is an integer greater than 2. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof, no proof by him has been found, his claim was discovered some 30 years after his death. This claim, which came to be known as Fermat's Last Theorem, stood unsolved for the next three and a half centuries; the claim became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics; the special case n = 4 - proved by Fermat himself - is sufficient to establish that if the theorem is false for some exponent n, not a prime number, it must be false for some smaller n, so only prime values of n need further investigation.
Over the next two centuries, the conjecture was proved for only the primes 3, 5, 7, although Sophie Germain innovated and proved an approach, relevant to an entire class of primes. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million, but a proof for all exponents was inaccessible. Separately, around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two different areas of mathematics. Known at the time as the Taniyama–Shimura–Weil conjecture, as the modularity theorem, it stood on its own, with no apparent connection to Fermat's Last Theorem, it was seen as significant and important in its own right, but was considered inaccessible to proof. In 1984, Gerhard Frey noticed an apparent link between these two unrelated and unsolved problems.
An outline suggesting this could be proved was given by Frey. The full proof that the two problems were linked was accomplished in 1986 by Ken Ribet, building on a partial proof by Jean-Pierre Serre, who proved all but one part known as the "epsilon conjecture"; these papers by Frey and Ribet showed that if the Modularity Theorem could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would follow automatically. The connection is described below: any solution that could contradict Fermat's Last Theorem could be used to contradict the Modularity Theorem. So if the modularity theorem were found to be true by definition no solution contradicting Fermat's Last Theorem could exist, which would therefore have to be true as well. Although both problems were daunting and considered to be "completely inaccessible" to proof at the time, this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers.
Important for researchers choosing a research topic was the fact that unlike Fermat's Last Theorem the Modularity Theorem was a major active research area for which a proof was desired and not just a historical oddity, so time spent working on it could be justified professionally. However, general opinion was that this showed the impracticality of proving the Taniyama–Shimura conjecture. Mathematician John Coates' quoted reaction was a common one: "I myself was sceptical that the beautiful link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture would lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to prove. I must confess I thought I wouldn’t see it proved in my lifetime." On hearing that Ribet had proven Frey's li
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1, not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers that are both smaller than 6. 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, unique up to their order; the property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and n. Faster algorithms include the Miller–Rabin primality test, fast but has a small chance of error, the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.
Fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number has 24,862,048 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled; the first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime numbers are still unsolved; these include Goldbach's conjecture, that every integer greater than 2 can be expressed as the sum of two primes, the twin prime conjecture, that there are infinitely many pairs of primes having just one 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. A natural number is called a prime number if it is greater than 1 and cannot be written as a product of two natural numbers that are both smaller than it; the numbers greater than 1 that are not prime are called composite numbers. In other words, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange n dots into a rectangular grid, more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, 5 are the prime numbers, as there are no other numbers that divide them evenly. 1 is not prime, as it is excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. The divisors of a natural number n are the numbers.
Every natural number has both 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 two positive divisors, 1 and the number itself. Yet another way to express the same thing is that a number n is prime if it is greater than one and if none of the numbers 2, 3, …, n − 1 divides n evenly; the first 25 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. No number n greater than 2 is prime because any such number can be expressed as the product 2 × n / 2. Therefore, every prime number other than 2 is an odd number, is called an odd prime; when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are and decimal numbers that end in 0 or 5 are divisible by 5; the set of all primes is sometimes denoted by P or by P.
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 proves the infinitude of primes and the fundamental theorem of arithmetic, shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes. Around 1000 AD, the Islamic mathematician Alhazen found Wilson's theorem, characterizing the prime numbers as the numbers n that evenly divide
53 is the natural number following 52 and preceding 54. It is the 16th prime number. Fifty-three is the 16th prime number, it is an Eisenstein prime, a Sophie Germain prime. The sum of the first 53 primes is 5830, divisible by 53, a property shared by few other numbers. 53 written in hexadecimal is 35, that is, the same characters used in the decimal representation, but reversed. Four additional multiples of 53 share this property: 371 = 17316, 5141 = 141516, 99481 = 1849916, 8520280 = 082025816. Apart from the trivial case of single-digit decimals, no other number has this property. 53 can not be expressed as the sum of its base-10 digits, making 53 a self number. 53 is the smallest prime number. The atomic number of iodine Messier object M53, a magnitude 8.5 globular cluster in the constellation Coma Berenices The New General Catalogue object NGC 53, a magnitude 12.6 barred spiral galaxy in the constellation Tucana Fifty-three is: The racing number of Herbie, a fictional Volkswagen Beetle with a mind of his own, first appearing in the 1968 film The Love Bug The code for international direct dial phone calls to Cuba "53 Days" is a northeastern USA rock band "53 Days" a novel by Georges Perec In How the Grinch Stole Christmas!, its animated TV special the Grinch says he's put up with the who's Christmas cheer for 53 years.
Fictional 53rd Precinct in the Bronx was found in the TV comedy "Car 54, Where Are You?" "53rd & 3rd" a song by the Ramones The number of Hail Mary beads on a standard, five decade Catholic Rosary. The number of bytes in an Asynchronous Transfer Mode packet. UDP and TCP port number for the Domain Name System protocol. 53-TET is a musical temperament that has a fifth, closer to pure than our current system. 53 More Things To Do In Zero Gravity is a book mentioned in The Hitchhiker's Guide to the Galaxy The maximum number of players on a National Football League roster Most points by a rookie in an NBA playoff game, by Philadelphia's Wilt Chamberlain, 1960 Most field goals, by Michael Jordan, 1992
89 is the natural number following 88 and preceding 90. 89 is: the 24th prime number, following 83 and preceding 97. A Chen prime. A Pythagorean prime; the smallest Sophie Germain prime to start a Cunningham chain of the first kind of six terms. An Eisenstein prime with no imaginary part and real part of the form 3n − 1. A Fibonacci number and thus a Fibonacci prime as well; the first few digits of its reciprocal coincide with the Fibonacci sequence due to the identity 1 89 = ∑ n = 1 ∞ F × 10 − = 0.011235955 …. A Markov number, appearing in solutions to the Markov Diophantine equation with other odd-indexed Fibonacci numbers. M89 is the 10th Mersenne prime. Although 89 is not a Lychrel number in base 10, it is unusual that it takes 24 iterations of the reverse and add process to reach a palindrome. Among the known non-Lychrel numbers in the first 10000 integers, no other number requires that many or more iterations; the palindrome reached is unusually large. Eighty-nine is: The atomic number of actinium.
Messier object a magnitude 11.5 elliptical galaxy in the constellation Virgo. The New General Catalogue object NGC 89, a magnitude 13.5 peculiar spiral galaxy in the constellation Phoenix and a member of Robert's Quartet. The Oklahoma Redhawks, an American minor league baseball team, were known as the Oklahoma 89ers; the number alludes to the Land Run of 1889, when the Unassigned Lands of Oklahoma were opened to white settlement. The team's home of Oklahoma City was founded during this event. In Rugby, an "89" or eight-nine move is a phase following a scrum, in which the number 8 catches the ball and transfers it to number 9; the Elite 89 Award is presented by the U. S. NCAA to the participant in each of the NCAA's 89 championship finals with the highest grade point average. 89, a 2017 film about a football match, between Liverpool and Arsenal in 1989. Eighty-nine is also: The designation of Interstate 89, a freeway that runs from New Hampshire to Vermont The designation of U. S. Route 89, a north-south highway that runs from Montana to Arizona The ISBN Group Identifier for books published in Korea Pop Song 89 A model of the Texas Instruments calculator TI-89 California Proposition 89, a 2006 California ballot initiative on campaign finance reform The title of a currently-unreleased song by Bon Iver The greatest number of verses in a chapter of a book of the Bible other than the Book of Psalms—specifically Numbers chapter 7.
The number of units of each colour in the board game Blokus The number of the French department Yonne Information Is Beautiful cites eighty-nine as one of the words censored on the Chinese internet. Hellin's law
Cryptography or cryptology is the practice and study of techniques for secure communication in the presence of third parties called adversaries. More cryptography is about constructing and analyzing protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, electrical engineering, communication science, physics. Applications of cryptography include electronic commerce, chip-based payment cards, digital currencies, computer passwords, military communications. Cryptography prior to the modern age was synonymous with encryption, the conversion of information from a readable state to apparent nonsense; the originator of an encrypted message shares the decoding technique only with intended recipients to preclude access from adversaries. The cryptography literature uses the names Alice for the sender, Bob for the intended recipient, Eve for the adversary. Since the development of rotor cipher machines in World War I and the advent of computers in World War II, the methods used to carry out cryptology have become complex and its application more widespread.
Modern cryptography is based on mathematical theory and computer science practice. It is theoretically possible to break such a system, but it is infeasible to do so by any known practical means; these schemes are therefore termed computationally secure. There exist information-theoretically secure schemes that provably cannot be broken with unlimited computing power—an example is the one-time pad—but these schemes are more difficult to use in practice than the best theoretically breakable but computationally secure mechanisms; the growth of cryptographic technology has raised a number of legal issues in the information age. Cryptography's potential for use as a tool for espionage and sedition has led many governments to classify it as a weapon and to limit or prohibit its use and export. In some jurisdictions where the use of cryptography is legal, laws permit investigators to compel the disclosure of encryption keys for documents relevant to an investigation. Cryptography plays a major role in digital rights management and copyright infringement of digital media.
The first use of the term cryptograph dates back to the 19th century—originating from The Gold-Bug, a novel by Edgar Allan Poe. Until modern times, cryptography referred exclusively to encryption, the process of converting ordinary information into unintelligible form. Decryption is the reverse, in other words, moving from the unintelligible ciphertext back to plaintext. A cipher is a pair of algorithms that create the reversing decryption; the detailed operation of a cipher is controlled both by the algorithm and in each instance by a "key". The key is a secret a short string of characters, needed to decrypt the ciphertext. Formally, a "cryptosystem" is the ordered list of elements of finite possible plaintexts, finite possible cyphertexts, finite possible keys, the encryption and decryption algorithms which correspond to each key. Keys are important both formally and in actual practice, as ciphers without variable keys can be trivially broken with only the knowledge of the cipher used and are therefore useless for most purposes.
Ciphers were used directly for encryption or decryption without additional procedures such as authentication or integrity checks. There are two kinds of cryptosystems: asymmetric. In symmetric systems the same key is used to decrypt a message. Data manipulation in symmetric systems is faster than asymmetric systems as they use shorter key lengths. Asymmetric systems use a public key to encrypt a private key to decrypt it. Use of asymmetric systems enhances the security of communication. Examples of asymmetric systems include RSA, ECC. Symmetric models include the used AES which replaced the older DES. In colloquial use, the term "code" is used to mean any method of encryption or concealment of meaning. However, in cryptography, code has a more specific meaning, it means the replacement of a unit of plaintext with a code word. Cryptanalysis is the term used for the study of methods for obtaining the meaning of encrypted information without access to the key required to do so; some use the terms cryptography and cryptology interchangeably in English, while others use cryptography to refer to the use and practice of cryptographic techniques and cryptology to refer to the combined study of cryptography and cryptanalysis.
English is more flexible than several other languages in which crypto
Marie-Sophie Germain was a French mathematician and philosopher. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's library including ones by Leonhard Euler and from correspondence with famous mathematicians such as Lagrange and Gauss. One of the pioneers of elasticity theory, she won the grand prize from the Paris Academy of Sciences for her essay on the subject, her work on Fermat's Last Theorem provided a foundation for mathematicians exploring the subject for hundreds of years after. Because of prejudice against her sex, she was unable to make a career out of mathematics, but she worked independently throughout her life. Before her death, Gauss had recommended that she be awarded an honorary degree, but that never occurred. On June 27, 1831, she died from breast cancer. At the centenary of her life, a street and a girls’ school were named after her; the Academy of Sciences established the Sophie Germain Prize in her honor.
Marie-Sophie Germain was born on April 1776, in Paris, France, in a house on Rue Saint-Denis. According to most sources, her father, Ambroise-François, was a wealthy silk merchant, though some believe he was a goldsmith. In 1789, he was elected as a representative of the bourgeoisie to the États-Généraux, which he saw change into the Constitutional Assembly, it is therefore assumed that Sophie witnessed many discussions between her father and his friends on politics and philosophy. Gray proposes. Marie-Sophie had one younger sister, named Angélique-Ambroise, one older sister, named Marie-Madeline, her mother was named Marie-Madeline, this plethora of "Maries" may have been the reason she went by Sophie. Germain's nephew Armand-Jacques Lherbette, Marie-Madeline's son, published some of Germain's work after she died; when Germain was 13, the Bastille fell, the revolutionary atmosphere of the city forced her to stay inside. For entertainment she turned to her father's library. Here she found J. E. Montucla's L'Histoire des Mathématiques, his story of the death of Archimedes intrigued her.
Sophie Germain thought that if the geometry method, which at that time referred to all of pure mathematics, could hold such fascination for Archimedes, it was a subject worthy of study. So she pored over every book on mathematics in her father's library teaching herself Latin and Greek so she could read works like those of Sir Isaac Newton and Leonhard Euler, she enjoyed Traité d'Arithmétique by Étienne Bézout and Le Calcul Différentiel by Jacques Antoine-Joseph. Her cousin visited Germain at home, encouraging her in her studies. Germain's parents did not at all approve of her sudden fascination with mathematics, thought inappropriate for a woman; when night came, they would deny her warm clothes and a fire for her bedroom to try to keep her from studying, but after they left she would take out candles, wrap herself in quilts and do mathematics. As Lynn Osen describes, when her parents found Sophie "asleep at her desk in the morning, the ink frozen in the ink horn and her slate covered with calculations," they realized that their daughter was serious and relented.
After some time, her mother secretly supported her. In 1794, when Germain was 18, the École Polytechnique opened; as a woman, Germain was barred from attending, but the new system of education made the "lecture notes available to all who asked." The new method required the students to "submit written observations." Germain obtained the lecture notes and began sending her work to Joseph Louis Lagrange, a faculty member. She used the name of a former student Monsieur Antoine-Auguste Le Blanc, "fearing," as she explained to Gauss, "the ridicule attached to a female scientist." When Lagrange saw the intelligence of M. Le Blanc, he requested a meeting, thus Sophie was forced to disclose her true identity. Lagrange did not mind that Germain was a woman, he became her mentor, he visited her in her home. Germain first became interested in number theory in 1798 when Adrien-Marie Legendre published Essai sur la théorie des nombres. After studying the work, she opened correspondence with him on number theory, elasticity.
Legendre showed some of Germain's work in the Supplément to his second edition of the Théorie des Nombres, where he calls it très ingénieuse. Germain's interest in number theory was renewed when she read Carl Friedrich Gauss' monumental work Disquisitiones Arithmeticae. After three years of working through the exercises and trying her own proofs for some of the theorems, she wrote, again under the pseudonym of M. LeBlanc, to the author himself, one year younger than her; the first letter, dated 21 November 1804, discussed Gauss' Disquisitiones and presented some of Germain's work on Fermat's Last Theorem. In the letter, Germain claimed to have proved the theorem for n = p – 1, where p is a prime number of the form p = 8k + 7. However, her proof contained a weak assumption, Gauss' reply did not comment on Germain's proof. Around 1807, during the Napoleonic wars, the French were occupying the German town of Braunschweig, where Gauss lived. Germain, concerned that he might suffer the fate of Archimedes, wrote to General Pernety, a family friend, requesting that he ensure Gauss' safety.
General Pernety sent a chief of a battalion to meet with Gauss to see that he was safe. As it turned out, Gauss was fine, but he was confused by the menti
In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions. Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is always true, rather than enumerate many confirmatory cases. An unproved proposition, believed to be true is known as a conjecture. Proofs employ logic but include some amount of natural language which admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory; the distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, so-called folk mathematics.
The philosophy of mathematics is concerned with the role of language and logic in proofs, mathematics as a language. The word "proof" comes from the Latin probare meaning "to test". Related modern words are the English "probe", "probation", "probability", the Spanish probar, Italian provare, the German probieren; the early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony. Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof, it is that the idea of demonstrating a conclusion first arose in connection with geometry, which meant the same as "land measurement". The development of mathematical proof is the product of ancient Greek mathematics, one of the greatest achievements thereof. Thales and Hippocrates of Chios proved some theorems in geometry. Eudoxus and Theaetetus formulated did not prove them.
Aristotle said definitions should describe the concept being defined in terms of other concepts known. Mathematical proofs were revolutionized by Euclid, who introduced the axiomatic method still in use today, starting with undefined terms and axioms, used these to prove theorems using deductive logic, his book, the Elements, was read by anyone, considered educated in the West until the middle of the 20th century. In addition to theorems of geometry, such as the Pythagorean theorem, the Elements covers number theory, including a proof that the square root of two is irrational and that there are infinitely many prime numbers. Further advances took place in medieval Islamic mathematics. While earlier Greek proofs were geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, etc. for "lines."
He used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle. Alhazen developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption; as practiced, a proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a statement. The standard of rigor has varied throughout history. A proof can be presented differently depending on the intended audience. In order to gain acceptance, a proof has to meet communal statements of rigor; the concept of a proof is formalized in the field of mathematical logic. A formal proof is written in a formal language instead of a natural language. A formal proof is defined as sequence of formulas in a formal language, in which each formula is a logical consequence of preceding formulas.
Having a definition of formal proof makes the concept of proof amenable to study. Indeed, the field of proof theory studies formal proofs and their properties, for example, the property that a statement has a formal proof. An application of proof theory is to show; the definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated proof assistants, this is done in practice. A classic question in philosophy a