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Heegner Number
In number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ..., a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, the ring of algebraic integers of \Q\left sqrt\right/math> has unique factorization. The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory. According to the (Baker–) Stark–Heegner theorem there are precisely nine Heegner numbers: This result was conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966, and Stark further indicated that the gap in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factorization, polynomial factoring, number theory, and algebraic geometry. The discriminant of the quadratic polynomial ax^2+bx+c is :b^2-4ac, the quantity which appears under the square root in the quadratic formula. If a\ne 0, this discriminant is zero if and only if the polynomial has a double root. In the case of real number, real coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct complex conjugate roots. Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
J-invariant
In mathematics, Felix Klein's -invariant or function is a modular function of weight zero for the special linear group \operatorname(2,\Z) defined on the upper half-plane of complex numbers. It is the unique such function that is holomorphic away from a simple pole at the cusp such that :j\big(e^\big) = 0, \quad j(i) = 1728 = 12^3. Rational functions of j are modular, and in fact give all modular functions of weight 0. Classically, the j-invariant was studied as a parameterization of elliptic curves over \mathbb, but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine). Definition The -invariant can be defined as a function on the upper half-plane \mathcal=\, by :j(\tau) = 1728 \frac = 1728 \frac = 1728 \frac with the third definition implying j(\tau) can be expressed as a cube, also since 1728 = 12^3. The function cannot be continued analytically beyond the upper half-plane due to the natura ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-expansion
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory. Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group \mathrm_2(\mathbb Z) \subset \mathrm_2(\mathbb R). Every modular form is attached to a Galois representation. The term "modular form", as a systematic description, is usually attributed to Erich Hecke. The importance of modular forms across multiple field of mathematics has been humorously represen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer Lattice (group), lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in algebraic number theory, allowing some features of the theory of cyclotomic fields to be carried over to wider areas of application. David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science. There is also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feynman Point
A sequence of six consecutive nines occurs in the decimal representation of the number pi (), starting at the 762nd decimal place.. It has become famous because of the mathematical coincidence, and because of the idea that one could memorize the digits of up to that point, and then suggest that is rational. The earliest known mention of this idea occurs in Douglas Hofstadter's 1985 book '' Metamagical Themas'', where Hofstadter states This sequence of six nines is colloquially known as the "Feynman point", after physicist Richard Feynman, who allegedly stated this same idea in a lecture.. However it is not clear when, or even if, Feynman ever made such a statement. It is not mentioned in his memoirs and unknown to his biographer James Gleick. Related statistics is conjectured, but not known, to be a normal number. For a normal number sampled uniformly at random, the probability of a specific sequence of six digits occurring this early in the decimal representation is about ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Srinivasa Ramanujan
Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable. Ramanujan initially developed his own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered". Seeking mathematicians who could better understand his work, in 1913 he began a mail correspondence with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognising Ramanujan's work as extraordinary, Hardy arranged ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing magic, scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lewis Carroll, L. Frank Baum, and G. K. Chesterton.Martin (2010) He was a leading authority on Lewis Carroll; '' The Annotated Alice'', which incorporated the text of Carroll's two Alice books, was his most successful work and sold over a million copies.Martin Gardner obituary (2010) He had a lifelong interest in magic and illusion and in 1999, ''MAGIC'' magazine named him as one of the "10 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it, with more than 150 Nobel Prize-winners being featured since its inception. In print since 1845, it is the oldest continuously published magazine in the United States. ''Scientific American'' is owned by Springer Nature, which is a subsidiary of Holtzbrinck Publishing Group. History ''Scientific American'' was founded by inventor and publisher Rufus Porter (painter), Rufus Porter in 1845 as a four-page weekly newspaper. The first issue of the large-format New York City newspaper was released on August 28, 1845. Throughout its early years, much emphasis was placed on reports of what was going on at the United States Patent and Trademark Office, U.S. Patent Office. It also reported on a broad range of inventions including perpetual motion machines, an 1860 devi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
April Fools' Day
April Fools' Day or April Fool's Day (rarely called All Fools' Day) is an annual custom on the 1st of April consisting of practical jokes, hoaxes, and pranks. Jokesters often expose their actions by shouting "April Fool[s]!" at the recipient. Mass media can be involved with these pranks, which may be revealed as such the following day. The custom of setting aside a day for playing harmless pranks upon one's neighbor has been relatively common in the world historically. Origins Although many theories have been proposed throughout the years, the origin of April Fools' Day is not exactly known. A disputed association between 1 April and foolishness is in Geoffrey Chaucer's ''The Canterbury Tales'' (1392). In the "Nun's Priest's Tale", a vain cock, Chauntecleer, is tricked by a fox "Since March began, full thirty days and two," i.e. the 32nd day from 1 March, which is 1 April. However, it is not clear that Chaucer was referencing 1 April since the text of the "Nun's Priest's Tale" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Hermite
Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermite polynomials, Hermite interpolation, Hermite normal form, Hermitian operators, and cubic Hermite splines are named in his honor. One of his students was Henri Poincaré. He was the first to prove that '' e'', the base of natural logarithms, is a transcendental number. His methods were used later by Ferdinand von Lindemann to prove that is transcendental. Life Hermite was born in Dieuze, Moselle, on 24 December 1822, with a deformity in his right foot that would impair his gait throughout his life. He was the sixth of seven children of Ferdinand Hermite and his wife, Madeleine née Lallemand. Ferdinand worked in the drapery business of Madeleine's family while also pursuing a career as an artist. The drapery business relocated to Nan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Integer
In recreational mathematics, an almost integer (or near-integer) is any number that is not an integer but is very close to one. Almost integers may be considered interesting when they arise in some context in which they are unexpected. Almost integers relating to the golden ratio and Fibonacci numbers Some examples of almost integers are high powers of the golden ratio \phi=\frac\approx 1.618, for example: : \begin \phi^ & =\frac\approx 3571.00028 \\ pt\phi^ & =2889+1292\sqrt5 \approx 5777.999827 \\ pt\phi^ & =\frac\approx 9349.000107 \end The fact that these powers approach integers is non-coincidental, because the golden ratio is a Pisot–Vijayaraghavan number. The ratios of Fibonacci or Lucas numbers can also make almost integers, for instance: * \frac \approx 1242282009792667284144565908481.999999999999999999999999999999195 * \frac \approx 2010054515457065378082322433761.000000000000000000000000000000497 The above examples can be generalized by the following sequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
