1.
Algebraic number theory
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Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of objects such as algebraic number fields and their rings of integers, finite fields. 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. Solutions to linear Diophantine equations, such as 26x + 65y =13, diophantus major work was the Arithmetica, of which only a portion has survived. Fermats last theorem was first conjectured by Pierre de Fermat in 1637, 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 number theory in the 19th century. In this book Gauss brings together results in number theory obtained by such as Fermat, Euler, Lagrange and Legendre. Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems, Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways. The Disquisitiones was the point for the work of other nineteenth century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet. Many of the annotations given by Gauss are in effect announcements of further research of his own and they must have appeared particularly cryptic to his contemporaries, we can now read them as containing the germs of the theories of L-functions and complex multiplication, in particular. In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, the formula, which Jacobi called a result touching the utmost of human acumen, opened the way for similar results regarding more general number fields. Based on his research of the structure of the group of quadratic fields, he proved the Dirichlet unit theorem. He first used the principle, a basic counting argument, in the proof of a theorem in diophantine approximation. He published important contributions to Fermats last theorem, for which he proved the cases n =5 and n =14, the Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers. Richard Dedekinds study of Lejeune Dirichlets work was what led him to his study of algebraic number fields. 1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory, Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, ideals generalize Ernst Eduard Kummers ideal numbers, devised as part of Kummers 1843 attempt to prove Fermats Last Theorem. David Hilbert unified the field of number theory with his 1897 treatise Zahlbericht

2.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n

3.
Elliptic curve
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In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form y 2 = x 3 + a x + b that is non-singular, that is, it has no cusps or self-intersections. Formally, a curve is a smooth, projective, algebraic curve of genus one. An elliptic curve is in fact an abelian variety – that is, it has a multiplication defined algebraically, often the curve itself, without O specified, is called an elliptic curve. The point O is actually the point at infinity in the projective plane, if y2 = P, where P is any polynomial of degree three in x with no repeated roots, then we obtain a nonsingular plane curve of genus one, which is thus an elliptic curve. If P has degree four and is square-free this equation describes a plane curve of genus one, however. Using the theory of functions, it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane. The torus is also a group, and in fact this correspondence is also a group isomorphism. Elliptic curves are important in number theory, and constitute a major area of current research, for example, they were used in the proof, by Andrew Wiles. They also find applications in elliptic curve cryptography and integer factorization, an elliptic curve is not an ellipse, see elliptic integral for the origin of the term. Topologically, an elliptic curve is a torus. In this context, a curve is a plane curve defined by an equation of the form y 2 = x 3 + a x + b where a and b are real numbers. This type of equation is called a Weierstrass equation, the definition of elliptic curve also requires that the curve be non-singular. Geometrically, this means that the graph has no cusps, self-intersections, algebraically, this involves calculating the discriminant Δ = −16 The curve is non-singular if and only if the discriminant is not equal to zero. The graph of a curve has two components if its discriminant is positive, and one component if it is negative. For example, in the shown in figure to the right, the discriminant in the first case is 64. When working in the plane, we can define a group structure on any smooth cubic curve. In Weierstrass normal form, such a curve will have a point at infinity, O. Since the curve is symmetrical about the x-axis, given any point P, if P and Q are two points on the curve, then we can uniquely describe a third point, P + Q, in the following way

4.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0

5.
Abelian variety
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Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field, historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a projective space. Abelian varieties defined over number fields are a special case. Localization techniques lead naturally from abelian varieties defined over fields to ones defined over finite fields. This induces a map from the field to any such finite field. Abelian varieties appear naturally as Jacobian varieties and Albanese varieties of other algebraic varieties, the group law of an abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve is a variety of dimension 1. Abelian varieties have Kodaira dimension 0, in the early nineteenth century, the theory of elliptic functions succeeded in giving a basis for the theory of elliptic integrals, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials, when those were replaced by polynomials of higher degree, say quintics, what would happen. In the work of Niels Abel and Carl Jacobi, the answer was formulated and this gave the first glimpse of an abelian variety of dimension 2, what would now be called the Jacobian of a hyperelliptic curve of genus 2. After Abel and Jacobi, some of the most important contributors to the theory of functions were Riemann, Weierstrass, Frobenius, Poincaré. The subject was popular at the time, already having a large literature. By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions, eventually, in the 1920s, Lefschetz laid the basis for the study of abelian functions in terms of complex tori. He also appears to be the first to use the name abelian variety and it was André Weil in the 1940s who gave the subject its modern foundations in the language of algebraic geometry. Today, abelian varieties form an important tool in number theory, in dynamical systems, a complex torus of dimension g is a torus of real dimension 2g that carries the structure of a complex manifold. It can always be obtained as the quotient of a complex vector space by a lattice of rank 2g. A complex abelian variety of dimension g is a torus of dimension g that is also a projective algebraic variety over the field of complex numbers

6.
Noam Elkies
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Noam David Elkies is an American mathematician and chess master. Along with A. O. L. Atkin, he extended Schoofs algorithm to create the Schoof–Elkies–Atkin algorithm, in 1993, when he was 26 years old, he became the youngest full professor in the history of Harvard University. He was a Putnam Fellow two more times during his undergraduate years, in 1987, he proved that an elliptic curve over the rational numbers is supersingular at infinitely many primes. In 1988, he found a counterexample to Eulers sum of powers conjecture for fourth powers and his work on these and other problems won him recognition and a position as an associate professor at Harvard in 1990. In 1993, he was made a full, tenured professor at the age of 26 and this made him the youngest full professor in the history of Harvard. Elkies, along with A. O. L. Atkin, in 1994 he was an invited speaker at the International Congress of Mathematicians in Zurich. In 2004 he received a Lester R. Ford Award, Elkies also studies the connections between music and mathematics. He sits on the Advisory Board of the Journal of Mathematics and he has discovered many new patterns in Conways Game of Life and has studied the mathematics of still life patterns in that cellular automaton rule. Elkies is a fellow at Harvards Lowell House, Elkies is a composer and solver of chess problems. He holds the title of National Master from the United States Chess Federation, but he no longer plays competitively

7.
Serge Lang
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Serge Lang was a French-born American mathematician and activist. He is known for his work in theory and for his mathematics textbooks. He was a member of the Bourbaki group, at the time of his death he was professor emeritus of mathematics at Yale University. Serge Lang was born in Saint-Germain-en-Laye close to Paris in 1927, Serge had a twin brother who became a basketball coach and a sister who became an actress. Lang moved with his family to California as a teenager, where he graduated in 1943 from Beverly Hills High School and he subsequently graduated from the California Institute of Technology in 1946, and received a doctorate from Princeton University in 1951. He held faculty positions at the University of Chicago, Columbia University, Lang studied under Emil Artin at Princeton University, writing his thesis on quasi-algebraic closure. Lang then worked on the geometric analogues of class field theory, later he moved into diophantine approximation and transcendental number theory, proving the Schneider–Lang theorem. A break in research while he was involved in trying to meet 1960s student activism halfway caused him difficulties in picking up the threads afterwards and he wrote on modular forms and modular units, the idea of a distribution on a profinite group, and value distribution theory. He introduced the Lang map and the Lang–Steinberg theorem in algebraic groups and he introduced the Katz–Lang finiteness theorem. He was a writer of mathematical texts, often completing one on his summer vacation. Most are at the graduate level and he wrote calculus texts and also prepared a book on group cohomology for Bourbaki. Langs Algebra, an introduction to abstract algebra, was a highly influential text that ran through numerous updated editions. His Steele prize citation stated, Langs Algebra changed the way graduate algebra is taught. It has affected all subsequent graduate-level algebra books, Lang was noted for his eagerness for contact with students. Many of his students at Yale considered him to be one of the greatest teachers of mathematics in the world and he won a Leroy P. Steele Prize for Mathematical Exposition from the American Mathematical Society. In 1960, he won the sixth Frank Nelson Cole Prize in Algebra for his paper Unramified class field theory over function fields in several variables, Lang spent much of his professional time engaged in political activism. He was a staunch socialist and active in opposition to the Vietnam War, Lang later quit his position at Columbia in 1971 in protest over the universitys treatment of anti-war protesters. Lang engaged in efforts to challenge anyone he believed was spreading misinformation or misusing science or mathematics to further their own goals. He attacked the 1977 Survey of the American Professoriate, a questionnaire that Seymour Martin Lipset

8.
Springer Science+Business Media
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Springer also hosts a number of scientific databases, including SpringerLink, Springer Protocols, and SpringerImages. Book publications include major works, textbooks, monographs and book series. Springer has major offices in Berlin, Heidelberg, Dordrecht, on 15 January 2015, Holtzbrinck Publishing Group / Nature Publishing Group and Springer Science+Business Media announced a merger. In 1964, Springer expanded its business internationally, opening an office in New York City, offices in Tokyo, Paris, Milan, Hong Kong, and Delhi soon followed. The academic publishing company BertelsmannSpringer was formed after Bertelsmann bought a majority stake in Springer-Verlag in 1999, the British investment groups Cinven and Candover bought BertelsmannSpringer from Bertelsmann in 2003. They merged the company in 2004 with the Dutch publisher Kluwer Academic Publishers which they bought from Wolters Kluwer in 2002, Springer acquired the open-access publisher BioMed Central in October 2008 for an undisclosed amount. In 2009, Cinven and Candover sold Springer to two private equity firms, EQT Partners and Government of Singapore Investment Corporation, the closing of the sale was confirmed in February 2010 after the competition authorities in the USA and in Europe approved the transfer. In 2011, Springer acquired Pharma Marketing and Publishing Services from Wolters Kluwer, in 2013, the London-based private equity firm BC Partners acquired a majority stake in Springer from EQT and GIC for $4.4 billion. In 2014, it was revealed that Springer had published 16 fake papers in its journals that had been computer-generated using SCIgen, Springer subsequently removed all the papers from these journals. IEEE had also done the thing by removing more than 100 fake papers from its conference proceedings. In 2015, Springer retracted 64 of the papers it had published after it was found that they had gone through a fraudulent peer review process, Springer provides its electronic book and journal content on its SpringerLink site, which launched in 1996. SpringerProtocols is home to a collection of protocols, recipes which provide step-by-step instructions for conducting experiments in research labs, SpringerImages was launched in 2008 and offers a collection of currently 1.8 million images spanning science, technology, and medicine. SpringerMaterials was launched in 2009 and is a platform for accessing the Landolt-Börnstein database of research and information on materials, authorMapper is a free online tool for visualizing scientific research that enables document discovery based on author locations and geographic maps. The tool helps users explore patterns in scientific research, identify trends, discover collaborative relationships. While open-access publishing typically requires the author to pay a fee for copyright retention, for example, a national institution in Poland allows authors to publish in open-access journals without incurring any personal cost - but using public funds. Springer is a member of the Open Access Scholarly Publishers Association, the Academic Publishing Industry, A Story of Merger and Acquisition – via Northern Illinois University

9.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker

10.
American Mathematical Society
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The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. It was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, john Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, the result was the Bulletin of the New York Mathematical Society, with Fiske as editor-in-chief. The de facto journal, as intended, was influential in increasing membership, the popularity of the Bulletin soon led to Transactions of the American Mathematical Society and Proceedings of the American Mathematical Society, which were also de facto journals. In 1891 Charlotte Scott became the first woman to join the society, the society reorganized under its present name and became a national society in 1894, and that year Scott served as the first woman on the first Council of the American Mathematical Society. In 1951, the headquarters moved from New York City to Providence. The society later added an office in Ann Arbor, Michigan in 1984, in 1954 the society called for the creation of a new teaching degree, a Doctor of Arts in Mathematics, similar to a PhD but without a research thesis. Mary W. Gray challenged that situation by sitting in on the Council meeting in Atlantic City, when she was told she had to leave, she refused saying she would wait until the police came. After that time, Council meetings were open to observers and the process of democratization of the Society had begun, julia Robinson was the first female president of the American Mathematical Society but was unable to complete her term as she was suffering from leukemia. In 1988 the Journal of the American Mathematical Society was created, the 2013 Joint Mathematics Meeting in San Diego drew over 6,600 attendees. Each of the four sections of the AMS hold meetings in the spring. The society also co-sponsors meetings with other mathematical societies. The AMS selects a class of Fellows who have made outstanding contributions to the advancement of mathematics. The AMS publishes Mathematical Reviews, a database of reviews of mathematical publications, various journals, in 1997 the AMS acquired the Chelsea Publishing Company, which it continues to use as an imprint. Blogs, Blog on Blogs e-Mentoring Network in the Mathematical Sciences AMS Graduate Student Blog PhD + Epsilon On the Market Some prizes are awarded jointly with other mathematical organizations. The AMS is led by the President, who is elected for a two-year term, morrey, Jr. Oscar Zariski Nathan Jacobson Saunders Mac Lane Lipman Bers R. H. Andrews Eric M. Friedlander David Vogan Robert L

11.
Mersenne prime
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In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a number that can be written in the form Mn = 2n −1 for some integer n. They are named after Marin Mersenne, a French Minim friar, the first four Mersenne primes are 3,7,31, and 127. If n is a number then so is 2n −1. The definition is therefore unchanged when written Mp = 2p −1 where p is assumed prime, more generally, numbers of the form Mn = 2n −1 without the primality requirement are called Mersenne numbers. The smallest composite pernicious Mersenne number is 211 −1 =2047 =23 ×89, Mersenne primes Mp are also noteworthy due to their connection to perfect numbers. As of January 2016,49 Mersenne primes are known, the largest known prime number 274,207,281 −1 is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search”, many fundamental questions about Mersenne primes remain unresolved. It is not even whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes,23 | M11,47 | M23,167 | M83,263 | M131,359 | M179,383 | M191,479 | M239, and 503 | M251. Since for these primes p, 2p +1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p +1, since p is a prime, it must be p or 1. The first four Mersenne primes are M2 =3, M3 =7, M5 =31, a basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity 2 a b −1 = ⋅ = ⋅ and this rules out primality for Mersenne numbers with composite exponent, such as M4 =24 −1 =15 =3 ×5 = ×. Though the above examples might suggest that Mp is prime for all p, this is not the case. The evidence at hand does suggest that a randomly selected Mersenne number is more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime Mp appear to grow increasingly sparse as p increases, in fact, of the 2,270,720 prime numbers p up to 37,156,667, Mp is prime for only 45 of them. The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, the Lucas–Lehmer primality test is an efficient primality test that greatly aids this task. The search for the largest known prime has somewhat of a cult following, consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing

12.
Pythagorean prime
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A Pythagorean prime is a prime number of the form 4n +1. Pythagorean primes are exactly the odd numbers that are the sum of two squares. For instance, the number 5 is a Pythagorean prime, √5 is the hypotenuse of a triangle with legs 1 and 2. The first few Pythagorean primes are 5,13,17,29,37,41,53,61,73,89,97,101,109,113, by Dirichlets theorem on arithmetic progressions, this sequence is infinite. More strongly, for n, the numbers of Pythagorean and non-Pythagorean primes up to n are approximately equal. However, the number of Pythagorean primes up to n is frequently smaller than the number of non-Pythagorean primes. For example, the values of n up to 600000 for which there are more Pythagorean than non-Pythagorean odd primes are 26861 and 26862. Sum of one odd square and one square is congruent to 1 mod 4. Fermats theorem on sums of two states that the prime numbers that can be represented as sums of two squares are exactly 2 and the odd primes congruent to 1 mod 4. The representation of such number is unique, up to the ordering of the two squares. Another way to understand this representation as a sum of two squares involves Gaussian integers, the numbers whose real part and imaginary part are both integers. The norm of a Gaussian integer x + yi is the number x2 + y2, thus, the Pythagorean primes occur as norms of Gaussian integers, while other primes do not. Within the Gaussian integers, the Pythagorean primes are not considered to be prime numbers, similarly, their squares can be factored in a different way than their integer factorization, as p2 =22 =. The real and imaginary parts of the factors in these factorizations are the leg lengths of the right triangles having the given hypotenuses, in the finite field Z/p with p a Pythagorean prime, the polynomial equation x2 = −1 has two solutions. This may be expressed by saying that −1 is a quadratic residue mod p, in contrast, this equation has no solution in the finite fields Z/p where p is an odd prime but is not Pythagorean. Pythagorean Primes, including 5,13 and 137, sloanes A007350, Where prime race 4n-1 vs. 4n+1 changes leader. The On-Line Encyclopedia of Integer Sequences

13.
Pierpont prime
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A Pierpont prime is a prime number of the form 2 u 3 v +1 for some nonnegative integers u and v. That is, they are the prime numbers p for which p −1 is 3-smooth. They are named after the mathematician James Pierpont, who introduced them in the study of regular polygons that can be constructed using conic sections. It is possible to prove that if v =0 and u >0, then u must be a power of 2, if v is positive then u must also be positive, and the Pierpont prime is of the form 6k +1. Empirically, the Pierpont primes do not seem to be rare or sparsely distributed. There are 36 Pierpont primes less than 106,59 less than 109,151 less than 1020, there are few restrictions from algebraic factorisations on the Pierpont primes, so there are no requirements like the Mersenne prime condition that the exponent must be prime. As there are Θ numbers of the form in this range. Andrew M. Gleason made this explicit, conjecturing there are infinitely many Pierpont primes. According to Gleasons conjecture there are Θ Pierpont primes smaller than N, when 2 u >3 v, the primality of 2 u 3 v +1 can be tested by Proths theorem. As part of the ongoing search for factors of Fermat numbers. The following table gives values of m, k, and n such that k ⋅2 n +1 divides 22 m +1, the left-hand side is a Pierpont prime when k is a power of 3, the right-hand side is a Fermat number. As of 2017, the largest known Pierpont prime is 3 ×210829346 +1, whose primality was discovered by Sai Yik Tang, in the mathematics of paper folding, the Huzita axioms define six of the seven types of fold possible. It has been shown that these folds are sufficient to allow the construction of the points that solve any cubic equation. It follows that they allow any regular polygon of N sides to be formed, as long as N >3 and of the form 2m3nρ and this is the same class of regular polygons as those that can be constructed with a compass, straightedge, and angle-trisector. Regular polygons which can be constructed with compass and straightedge are the special case where n =0 and ρ is a product of distinct Fermat primes, themselves a subset of Pierpont primes. In 1895, James Pierpont studied the same class of regular polygons, Pierpont generalized compass and straightedge constructions in a different way, by adding the ability to draw conic sections whose coefficients come from previously constructed points. As he showed, the regular N-gons that can be constructed with these operations are the ones such that the totient of N is 3-smooth. Since the totient of a prime is formed by subtracting one from it, however, Pierpont did not describe the form of the composite numbers with 3-smooth totients. As Gleason later showed, these numbers are exactly the ones of the form 2m3nρ given above, the smallest prime that is not a Pierpont prime is 11, therefore, the hendecagon is the smallest regular polygon that cannot be constructed with compass, straightedge and angle trisector

14.
Pell number
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In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1,2,5,12, and 29. The numerators of the sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers, these numbers form a second infinite sequence that begins with 2,6,14,34. As with Pells equation, the name of the Pell numbers stems from Leonhard Eulers mistaken attribution of the equation, the Pell–Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type, the Pell and companion Pell numbers are Lucas sequences. The Pell numbers are defined by the recurrence relation P n = {0 if n =0,1 if n =1,2 P n −1 + P n −2 otherwise. In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. The first few terms of the sequence are 0,1,2,5,12,29,70,169,408,985,2378,5741,13860, …. The Pell numbers can also be expressed by the closed form formula P n = n − n 22, a third definition is possible, from the matrix formula = n. Pell numbers arise historically and most notably in the rational approximation to √2. If two large integers x and y form a solution to the Pell equation x 2 −2 y 2 = ±1 and that is, the solutions have the form P n −1 + P n P n. The approximation 2 ≈577408 of this type was known to Indian mathematicians in the third or fourth century B. C, the Greek mathematicians of the fifth century B. C. also knew of this sequence of approximations, Plato refers to the numerators as rational diameters. In the 2nd century CE Theon of Smyrna used the term the side and these approximations can be derived from the continued fraction expansion of 2,2 =1 +12 +12 +12 +12 +12 + ⋱. As Knuth describes, the fact that Pell numbers approximate √2 allows them to be used for accurate rational approximations to an octagon with vertex coordinates. All vertices are equally distant from the origin, and form uniform angles around the origin. Alternatively, the points, and form approximate octagons in which the vertices are equally distant from the origin. A Pell prime is a Pell number that is prime, the first few Pell primes are 2,5,29,5741, …. The indices of these primes within the sequence of all Pell numbers are 2,3,5,11,13,29,41,53,59,89,97,101,167,181,191, … These indices are all themselves prime. As with the Fibonacci numbers, a Pell number Pn can only be prime if n itself is prime, the only Pell numbers that are squares, cubes, or any higher power of an integer are 0,1, and 169 =132. However, despite having so few squares or other powers, Pell numbers have a connection to square triangular numbers

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Partition (number theory)
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In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition, a summand in a partition is also called a part. The number of partitions of n is given by the function p. The notation λ ⊢ n means that λ is a partition of n, Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials, the symmetric group and in group representation theory in general. For example, the partition 2 +2 +1 might instead be written as the tuple or in the more compact form where the superscript indicates the number of repetitions of a term. There are two common methods to represent partitions, as Ferrers diagrams, named after Norman Macleod Ferrers. Both have several possible conventions, here, we use English notation, with diagrams aligned in the upper-left corner. The partition 6 +4 +3 +1 of the positive number 14 can be represented by the diagram, The 14 circles are lined up in 4 rows. The diagrams for the 5 partitions of the number 4 are listed below, rather than representing a partition with dots, as in the Ferrers diagram, the Young diagram uses boxes or squares. As a type of shape made by adjacent squares joined together, by convention p =1, p =0 for n negative. The first few values of the function are,1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,1002,1255,1575,1958,2436,3010,3718,4565,5604. As of June 2013, the largest known prime number that counts a number of partitions is p, the generating function for p is given by, ∑ n =0 ∞ p x n = ∏ k =1 ∞. Expanding each factor on the side as a geometric series. The xn term in this product counts the number of ways to write n = a1 + 2a2 + 3a3 +, where each number i appears ai times. This is precisely the definition of a partition of n, so our product is the generating function. More generally, the function for the partitions of n into numbers from a set A can be found by taking only those terms in the product where k is an element of A. This result is due to Euler, the formulation of Eulers generating function is a special case of a q-Pochhammer symbol and is similar to the product formulation of many modular forms, and specifically the Dedekind eta function