question: How do we know that the value for e and pi supercomputers compute are correct?
comment: you don’t just ask a computer “what are the first million digits of pi?” we’ve known since long before modern computers how to compute digits of pi. these super powerful computers just let us do it faster. we know they are correct because we told the computers how to do it
passages: Calculating digits of the decimal expansion of constants has been a common enterprise for many centuries. For example, German mathematician Ludolph van Ceulen of the 16th century spent a major part of his life calculating the first 35 digits of pi. Using computers and supercomputers, some of the mathematical constants, including π, "e", and the square root of 2, have been computed to more than one hundred billion digits. Fast algorithms have been developed, some of which — as for Apéry's constant — are unexpectedly fast.
   In 1989, the Chudnovsky brothers computed to over 1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan's infinite series of :
  
  In 1999, Yasumasa Kanada and his team at the University of Tokyo computed to over 200 billion decimal places on the supercomputer HITACHI SR8000/MPP (128 nodes) using another variation of Ramanujan's infinite series of . In October 2005, they claimed to have calculated it to 1.24 trillion places.
   The first one million digits of and are available from Project Gutenberg (see external links below). A former calculation record (December 2002) by Yasumasa Kanada of Tokyo University stood at 1.24 trillion digits, which were computed in September 2002 on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits). The following Machin-like formulæ were used for this:
   In August 2009, a Japanese supercomputer called the T2K Open Supercomputer more than doubled the previous record by calculating to roughly 2.6 trillion digits in approximately 73 hours and 36 minutes.
  
  In December 2009, Fabrice Bellard used a home computer to compute 2.7 trillion decimal digits of . Calculations were performed in base 2 (binary), then the result was converted to base 10 (decimal). The calculation, conversion, and verification steps took a total of 131 days.
   After developing these indices, one can calculate both PIT and TTC PDs for counterparties within each of the covered sectors. To obtain PIT PDs, one introduces the relevant indices into the relevant default models, re-calibrate the models to defaults, and apply the models with current and projected changes in indices as inputs. If a PD model weren’t otherwise PIT, the introduction of the indices will make it PIT. The specific model formulation depends on the features important to each, distinguished class of counterparties and data constraints. Some common approaches include:
   In addition to exact analytical expressions for representation of , there are stochastic techniques for estimating . One such approach begins with an infinite sequence of independent random variables , …, drawn from the uniform distribution on [0, 1]. Let be the least number such that the sum of the first observations exceeds 1:
  
  Then the expected value of is : .
  
  Section::::Representations.:Known digits.
  
  The number of known digits of has increased substantially during the last decades. This is due both to the increased performance of computers and to algorithmic improvements.
   One can estimate that the number of floating-point operations required to complete the simulation is proportional to the number of mesh points and the number of time steps, and in conclusion, the number of operations grows as formula_29.
   In April 2012, the factorization of formula_1 by a room temperature (300K) NMR adiabatic quantum computer was reported by a group led by Xinhua Peng. In November 2014 it was discovered by Nike Dattani and Nathan Bryans that the 2012 experiment had in fact also factored much larger numbers without knowing it. In April 2016 the 18-bit number 200099 was factored using quantum annealing on a D-Wave 2X quantum processor. Shortly after, 291311 was factored using NMR at higher than room temperature.. The current largest number factored on a quantum device is 4088459, which was achieved using IBM's 5-qubit processor.
   On December 12, 2009, a team including researchers from the CWI, the EPFL, INRIA and NTT in addition to the authors of the previous record factored RSA-768, a 232-digit semiprime. They used the equivalent of almost 2000
  
  years of computing on a single core 2.2 GHz AMD Opteron.
  
  Section::::Numbers of a special form.
  
  12 − 1, of 542 bits (163 digits), was factored between April and July 1993 by a team at CWI and Oregon State University.
   On May 9, 2005, F. Bahr, M. Boehm, J. Franke, and T. Kleinjung announced that they had factorized the number using GNFS as follows:
  
  The CPU time spent on finding these factors by a collection of parallel computers amounted – very approximately – to the equivalent of 75 years work for a single 2.2 GHz Opteron-based computer. Note that while this approximation serves to suggest the scale of the effort, it leaves out many complicating factors; the announcement states it more precisely.
  
  Section::::RSA-210.
  
  RSA-210 has 210 decimal digits (696 bits) and was factored in September 2013 by Ryan Propper:
  
  Section::::RSA-704.
   The record of manual approximation of is held by William Shanks, who calculated 527 digits correctly in the years preceding 1873. Since the middle of the 20th century, the approximation of has been the task of electronic digital computers; , the record was 22.4trillion digits. (For a comprehensive account, see Chronology of computation of.) In March 2019 Emma Haruka Iwao, a Google employee from Japan calculated to a new world record length of 31 trillion digits with the help of the company's cloud computing service.
  
  Section::::Early history.
   In 1996, Simon Plouffe derived an algorithm to extract the th decimal digit of (using base10 math to extract a base10 digit), and which can do so with an improved speed of time. The algorithm requires virtually no memory for the storage of an array or matrix so the one-millionth digit of can be computed using a pocket calculator. However, it would be quite tedious and impractical to do so.
  
  The calculation speed of Plouffe's formula was improved to by Fabrice Bellard, who derived an alternative formula (albeit only in base2 math) for computing .
  
  Section::::Efficient methods.
   In August 2010, Shigeru Kondo used Alexander Yee's y-cruncher to calculate 5 trillion digits of . This was the world record for any type of calculation, but significantly it was performed on a home computer built by Kondo. The calculation was done between 4 May and 3 August, with the primary and secondary verifications taking 64 and 66 hours respectively.
  
  In October 2011, Shigeru Kondo broke his own record by computing ten trillion (10) and fifty digits using the same method but with better hardware.
   In June 2011, France's Tera 100 was certified the fastest supercomputer in Europe, and ranked 9th in the world at the time. It was the first petascale supercomputer designed and built in Europe.
  
  Switzerland's Piz Daint is the fastest European supercomputer, following an upgrade in October 2016 it is ranked 3rd in the world with a peak of over 25 petaflops.
   BULLET::::- Hermes Project: C/C++/Python library for rapid prototyping of space- and space-time adaptive hp-FEM solvers for a large variety of PDEs and multiphysics PDE systems, developed by the hp-FEM group at the University of Nevada, Reno (USA), Institute of Thermomechanics, Prague (Czech Republic), and the University of West Bohemia in Pilsen (Czech Republic) – with the Agros2D engineering software built on top of the Hermes library.
   In April 2003, the same team factored RSA-160 using about a hundred CPUs at BSI, with the final stages of the calculation done using 25 processors of an SGI Origin supercomputer.
  
  The 174-digit RSA-576 was factored by Franke, Kleinjung and members of the NFSNET collaboration in December 2003, using resources at BSI and the University of Bonn; soon afterwards, Aoki, Kida, Shimoyama, Sonoda and Ueda announced that they had factored a 164-digit cofactor of 2+1.
   BULLET::::- 2007 – a team of researchers throughout North America and Europe used networks of computers to map E.
  
  BULLET::::- 2009 – Fundamental lemma (Langlands program) had been proved by Ngô Bảo Châu.
  
  BULLET::::- 2010 – Larry Guth and Nets Hawk Katz solve the Erdős distinct distances problem.
  
  BULLET::::- 2013 – Yitang Zhang proves the first finite bound on gaps between prime numbers.
  
  BULLET::::- 2014 – Project Flyspeck announces that it completed proof of Kepler's conjecture.
  
  BULLET::::- 2014 – Using Alexander Yee's y-cruncher "houkouonchi" successfully calculated π to 13.3 trillion digits.
  
  BULLET::::- 2015 – Terence Tao solved The Erdös Discrepancy Problem
   For instance, in the IPO filing for Google in 2004, rather than a typical round-number amount of money, the company announced its intention to raise $2,718,281,828, which is billion dollars rounded to the nearest dollar. Google was also responsible for a billboard that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and Austin, Texas. It read "{first 10-digit prime found in consecutive digits of }.com". Solving this problem and visiting the advertised (now defunct) web site led to an even more difficult problem to solve, which in turn led to Google Labs where the visitor was invited to submit a résumé. The first 10-digit prime in is 7427466391, which starts at the 99th digit.
   In December 2013, Kondo broke his own record for a second time when he computed 12.1 trillion digits of .
  
  In October 2014, Sandon Van Ness, going by the pseudonym "houkouonchi" used y-cruncher to calculate 13.3 trillion digits of .
  
  In November 2016, Peter Trueb and his sponsors computed on y-cruncher and fully verified 22.4 trillion digits of . The computation took (with three interruptions) 105 days to complete.
  
  In March 2019, Emma Haruka Iwao, an employee at Google, computed 31.4 trillion digits of pi using y-cruncher and Google Cloud machines. This took 121 days to complete.
  
  Section::::Practical approximations.
   As an example for SPECint2006, consider a processor which can run 400.perlbench in 2000 seconds. The time it takes the reference machine to run the benchmark is 9770 seconds. Thus the ratio is 4.885. Each ratio is computed, and then the geometric mean of those ratios is computed to produce an overall value.
  
  Section::::Background.
   In 1991 Victor R. Bond and Michael F. Fraietta created an efficient and highly accurate method for solving the two-body perturbed problem. This method uses the linearized and regularized differential equations of motion derived by Hans Sperling and a perturbation theory based on these equations developed by C.A. Burdet in the year 1864. In 1973, Bond and Hanssen improved Burdet's set of differential equations by using the total energy of the perturbed system as a parameter instead of the two-body energy and by reducing the number of elements to 13. In 1989 Bond and Gottlieb embedded the Jacobian integral, which is a constant when the potential function is explicitly dependent upon time as well as position in the Newtonian equations. The Jacobian constant was used as an element to replace the total energy in a reformulation of the differential equations of motion. In this process, another element which is proportional to a component of the angular momentum is introduced. This brought the total number of elements back to 14. In 1991, Bond and Fraietta made further revisions by replacing the Laplace vector with another vector integral as well as another scalar integral which removed small secular terms which appeared in the differential equations for some of the elements.
   By the 5th century CE, was known to about seven digits in Chinese mathematics, and to about five in Indian mathematics. Further progress was not made for nearly a millennium, until the 14th century, when Indian mathematician and astronomer Madhava of Sangamagrama, founder of the Kerala school of astronomy and mathematics, discovered the infinite series for , now known as the Madhava–Leibniz series, and gave two methods for computing the value of . One of these methods is to obtain a rapidly converging series by transforming the original infinite series of . By doing so, he obtained the infinite series
   Pi is a fundamental constant of nature. Archimedes discovered that the area of the circle equals the square of its radius times pi. Archimedes was the first to accurately calculate pi by using polygons with 96 sides both inside and outside a circle then measuring the line segments and finding that pi was between formula_5 and formula_6. A Chinese calculation used polygons with 3,000 sides and calculated pi accurately to five decimal places. The Chinese also found that formula_7 was an accurate estimate of pi to within 6 decimal places and was the most accurate estimate for 1,000 years until arabic numerals were used for arithmetic. 
   The next confirmed DES cracker was the COPACOBANA machine built in 2006 by teams of the Universities of Bochum and Kiel, both in Germany. Unlike the EFF machine, COPACOBANA consists of commercially available, reconfigurable integrated circuits. 120 of these field-programmable gate arrays (FPGAs) of type XILINX Spartan-3 1000 run in parallel. They are grouped in 20 DIMM modules, each containing 6 FPGAs. The use of reconfigurable hardware makes the machine applicable to other code breaking tasks as well. One of the more interesting aspects of COPACOBANA is its cost factor. One machine can be built for approximately $10,000. The cost decrease by roughly a factor of 25 over the EFF machine is an example of the continuous improvement of digital hardware—see Moore's law. Adjusting for inflation over 8 years yields an even higher improvement of about 30x. Since 2007, SciEngines GmbH, a spin-off company of the two project partners of COPACOBANA has enhanced and developed successors of COPACOBANA. In 2008 their COPACOBANA RIVYERA reduced the time to break DES to less than one day, using 128 Spartan-3 5000's. SciEngines RIVYERA held the record in brute-force breaking DES, having utilized 128 Spartan-3 5000 FPGAs. Their 256 Spartan-6 LX150 model has further lowered this time.
   Records since then have all been accomplished on personal computers using the Chudnovsky algorithm. In 2009, Fabrice Bellard computed just under 2.7 trillion digits, and from 2010 onward, all records have been set using Alexander Yee's y-cruncher software. , the record stands at 22,459,157,718,361 ( × 10) digits. The limitation on further expansion is primarily storage space for the computation.
  
  In November 2002, Yasumasa Kanada and a team of 9 others used the Hitachi SR8000, a 64-node supercomputer with 1 terabyte of main memory, to calculate to roughly 1.24 trillion digits in around 600 hours.
  
presuppositions: Without computers, we wouldn't know the value for e and pi. Supercomputers can compute e and pi by themselves.
corrections: We've long known since before modern computers how to calculate digits of pi. Humans taught supercomputers how to compute pi and e, so we know they are correct.
raw_presuppositions: Without computers, we wouldn't know the value for e and pi. Supercomputers can compute e and pi by themselves.
raw_corrections: We've long known since before modern computers how to calculate digits of pi. Humans taught supercomputers how to compute pi and e, so we know they are correct.