Similar formulas follow from for all positive integers. Borwein et al. Further sums are given in Ramanujan ,. Beeler et al. Equation 78 is derived from a modular identity of order 58, although a first derivation was not presented prior to Borwein and Borwein The above series both give.
Such series exist because of the rationality of various modular invariants. A class number field involves th degree algebraic integers of the constants , , and. Of all series consisting of only integer terms, the one gives the most numeric digits in the shortest period of time corresponds to the largest class number 1 discriminant of and was formulated by the Chudnovsky brothers The appearing here is the same one appearing in the fact that the Ramanujan constant is very nearly an integer.
Similarly, the factor comes from the j -function identity for. The series is given by. Borwein and Borwein ; Beck and Trott; Bailey et al. This series gives 14 digits accurately per term. The same equation in another form was given by the Chudnovsky brothers and is used by the Wolfram Language to calculate Vardi ; Wolfram Research ,.
The best formula for class number 2 largest discriminant is. Borwein and Borwein This series adds about 25 digits for each additional term. The fastest converging series for class number 3 corresponds to and gives digits per term.
The fastest converging class number 4 series corresponds to and is. This gives 50 digits per term. Borwein and Borwein have developed a general algorithm for generating such series for arbitrary class number. A complete listing of Ramanujan's series for found in his second and third notebooks is given by Berndt , pp.
These equations were first proved by Borwein and Borwein a, pp. Borwein and Borwein b, , proved other equations of this type, and Chudnovsky and Chudnovsky found similar equations for other transcendental constants Bailey et al.
Adamchik, V. Monthly , , Backhouse, N. Pancake Functions and Approximations to. Bailey, D. Experimental Mathematics in Action. Beck, G. Beckmann, P. A History of Pi, 3rd ed. New York: Dorset Press, Beeler, M. Item in Beeler, M. Berndt, B. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, Beukers, F.
Blatner, D. The Joy of Pi. New York: Walker, Boros, G. Cambridge, England: Cambridge University Press, Borwein, J. Mathematics by Experiment: Plausible Reasoning in the 21st Century.
Experimentation in Mathematics: Computational Paths to Discovery. New York: Wiley, a. Andrews, B. Berndt, and R. New York: Academic Press, pp. Monthly 96 , , Castellanos, D. This site uses Akismet to reduce spam. Learn how your comment data is processed. The list of linear algebra problems is available here. Enter your email address to subscribe to this blog and receive notifications of new posts by email. Email Address. Conditional Probability Problems about Die Rolling.
Linear Algebra. Ring theory. Happy New Year !! Here is the list of mathematical facts about the number that you can brag about Miscellaneous Reference. Find the largest prime number less than one million. Pi formula can be expressed as,. Example 1: Noah measured the perimeter of the circular section of pipe as 88 inches. Calculate the diameter of the same pipe using the pi formula.
To find: The diameter of the pipe. Example 2: The diameter of a circular park measures inches. Using Pi formula calculate how much distance have you covered if you walked exactly 1 round across its boundary. To find: Circumference of the park. Example 3: James measured the perimeter of the circle as 66 units and the diameter of the same circle is 21 units.
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