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Chudnovsky Algorithm
The Chudnovsky algorithm is a fast method for calculating the digits of , based on Ramanujan's formulae. Published by the Chudnovsky brothers in 1988, it was used to calculate to a billion decimal places. It was used in the world record calculations of 2.7 trillion digits of in December 2009, 10 trillion digits in October 2011, 22.4 trillion digits in November 2016, 31.4 trillion digits in September 2018–January 2019, 50 trillion digits on January 29, 2020, 62.8 trillion digits on August 14, 2021, 100 trillion digits on March 21, 2022, 105 trillion digits on March 14, 2024, and 202 trillion digits on June 28, 2024. Recently, the record was broken yet again on April 2nd 2025 with 300 trillion digits of pi. This was done through the usage of the algorithm on y-cruncher. Algorithm The algorithm is based on the negated Heegner number d = -163 , the ''j''-function j \left(\tfrac\right) = -640320^3, and on the following rapidly convergent generalized hypergeometric series: \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Hypergeometric Series
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials. Notation A hypergeometric series is formally defined as a power series :\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_ \beta_n z^n in which the ratio of successive coefficients is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borwein's Algorithm
Borwein's algorithm was devised by Jonathan and Peter Borwein to calculate the value of 1 / \pi. This and other algorithms can be found in the book ''Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity''. Ramanujan–Sato series These two are examples of a Ramanujan–Sato series. The related Chudnovsky algorithm uses a discriminant with class number 1. Class number 2 (1989) Start by setting : \begin A & = 212175710912 \sqrt + 1657145277365 \\ B & = 13773980892672 \sqrt + 107578229802750 \\ C & = \left(5280\left(236674+30303\sqrt\right)\right)^3 \end Then :\frac = 12\sum_^\infty \frac Each additional term of the partial sum yields approximately 25 digits. Class number 4 (1993) Start by setting : \begin A = & 63365028312971999585426220 \\ & + 28337702140800842046825600\sqrt \\ & + 384\sqrt \big(10891728551171178200467436212395209160385656017 \\ & + \left. 487092908657881022 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bailey–Borwein–Plouffe Formula
The Bailey–Borwein–Plouffe formula (BBP formula) is a formula for . It was discovered in 1995 by Simon Plouffe and is named after the authors of the article in which it was published, David H. Bailey, Peter Borwein, and Plouffe. The formula is: : \pi = \sum_^\left frac \left(\frac-\frac-\frac-\frac\right)\right/math> The BBP formula gives rise to a spigot algorithm for computing the ''n''th base-16 (hexadecimal) digit of (and therefore also the ''4n''th binary digit of ) without computing the preceding digits. This does ''not'' compute the ''n''th decimal digit of (i.e., in base 10). But another formula discovered by Plouffe in 2022 allows extracting the ''n''th digit of in decimal. BBP and BBP-inspired algorithms have been used in projects such as PiHex for calculating many digits of using distributed computing. The existence of this formula came as a surprise. It had been widely believed that computing the ''n''th digit of is just as hard as computing the first ''n'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Splitting
In mathematics, binary splitting is a technique for speeding up numerical evaluation of many types of series with rational terms. In particular, it can be used to evaluate hypergeometric series at rational points. Method Given a series :S(a,b) = \sum_^b \frac where ''pn'' and ''qn'' are integers, the goal of binary splitting is to compute integers ''P''(''a'', ''b'') and ''Q''(''a'', ''b'') such that :S(a,b) = \frac{Q(a,b)}. The splitting consists of setting ''m'' = ''a'' + ''b'')/2and recursively computing ''P''(''a'', ''b'') and ''Q''(''a'', ''b'') from ''P''(''a'', ''m''), ''P''(''m'', ''b''), ''Q''(''a'', ''m''), and ''Q''(''m'', ''b''). When ''a'' and ''b'' are sufficiently close, ''P''(''a'', ''b'') and ''Q''(''a'', ''b'') can be computed directly from ''pa...pb'' and ''qa...qb''. Comparison with other methods Binary splitting requires more memory than direct term-by-term summation, but is asymptotically faster since the sizes of all occurring subproducts are re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Complexity
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan–Sato Series
In mathematics, a Ramanujan–Sato series generalizes Ramanujan's pi formulas such as, :\frac = \frac \sum_^\infty \frac \frac to the form :\frac = \sum_^\infty s(k) \frac by using other well-defined sequences of integers s(k) obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients \tbinom, and A,B,C employing modular forms of higher levels. Ramanujan made the enigmatic remark that there were "corresponding theories", but it was only in 2012 that H. H. Chan and S. Cooper found a general approach that used the underlying modular congruence subgroup \Gamma_0(n), while G. Almkvist has experimentally found numerous other examples also with a general method using differential operators. Levels ''1–4A'' were given by Ramanujan (1914), level ''5'' by H. H. Chan and S. Cooper (2012), ''6A'' by Chan, Tanigawa, Yang, and Zudilin, ''6B'' by Sato (2002), ''6C'' by H. Chan, S. Chan, and Z. Liu (2004), ''6D'' by H. Chan and H. Verrill ... [...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] |
List Of Formulae Involving π
The following is a list of significant formulae involving the mathematical constant . Many of these formulae can be found in the article '' Pi'', or the article '' Approximations of ''. Euclidean geometry : \pi = \frac Cd = \frac C where is the circumference of a circle, is the diameter, and is the radius. More generally, : \pi=\frac where and are, respectively, the perimeter and the width of any curve of constant width. : A = \pi r^2 where is the area of a circle. More generally, : A = \pi ab where is the area enclosed by an ellipse with semi-major axis and semi-minor axis . : C=\frac\left(a_1^2-\sum_^\infty 2^(a_n^2-b_n^2)\right) where is the circumference of an ellipse with semi-major axis and semi-minor axis and a_n,b_n are the arithmetic and geometric iterations of \operatorname(a,b), the arithmetic-geometric mean of and with the initial values a_0=a and b_0=b. : A=4\pi r^2 where is the area between the witch of Agnesi and its asymptotic line; is the ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Y-cruncher
y-cruncher is a computer program for the calculation of some mathematical constant with theoretical accuracy limited only by computing time and available storage space. It was originally developed to calculate the Euler-Mascheroni constant ; the y is derived from it in the name. Since 2010, y-cruncher has been used for all record calculations of the number pi and other constants. The software is downloadable from the website of the developers for Microsoft Windows and Linux. It does not have a graphical interface, but works on the command line. Calculation options are selected or entered via the text menu, the results are saved as a file. Some popular uses of y-cruncher are running hardware benchmarks to measure performance of computer system. An example of such benchmark iHWBOT y-cruncher can also be used for stress-tests, as performed computations are sensitive to RAM errors and the program can automatically detect such errors. Development Alexander J. Yee started dev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
