Last update: August 12 2010
Ten decimal places of p are sufficient to give the circumference of the earth to a fraction of an inch, and thirty decimal places would give the circumference of the visible universe to a quantity imperceptible to the most powerful microscope. Simon Newcomb (1835-1909)
I myself have yielded to the temptation of calculating quantities that interested me to a greater number of decimals than could ever be required, and can testify to the pleasure of admiring a long row of figures in a new result and realizing what a veritable triumph of algebra and arithmetic they represent. James Whitbread Lee Glaisher (1848-1928)
The following table illustrates the various difficulties encountered to compute some famous Mathematical Constants with as many digits as possible. The computation of many decimal places of those constants has involved for each of them a few and sometimes a lot of mathematicians.
It's of interest to notice that the methods used for this purpose are numerous and cover a large panel of classical algorithms: iterative algorithms, series expansion, binary splitting, FFT, numerical quadratures, sieve, Newton's iteration, eigenvalues, ... Of course all those constants have not attracted as much interest as the celebrated constant p and therefore it only gives an idea of the computational complexity of each of them. Observe that the memory required to store so many digits becomes huge for the last constants (and especially in Kanada's, Daisuke's, Bellard's or Yee-Kondo's latest impressive computation of the constant p) on a super PC.
Other interesting constants may be included in this table and the current records will be updated when necessary. To avoid mistakes a record must be established by two, as independent as possible, methods.
First digits |
Computed digits |
Who - Year | |
Brun's constant | 1.902160582... | 9 | T. Nicely - 1999 & P. Sebah - 2002 |
Gauss-Kuzmin-Wirsing | 0.30366300289873265... | 468 | K. Briggs - 2003 |
Artin's constant | 0.37395581361920228... | 1,000 | G. Niklasch - 1999 |
Fransén-Robinson | 2.80777024202851936... | 1,025 | P. Sebah - 2001 |
Twin prime constant | 0.66016181584686957... | 5,020 | P. Sebah - 2001 |
Mertens' constant | 0.26149721284764278... | 8,010 | P. Sebah - 2001 |
Landau-Ramanujan K | 0.76422365358922066... | 30,010 | P. Sebah - 2002 |
Soldner-Ramanujan |
1.45136923488338105... |
75,500 | P. Sebah - 2001 |
Khintchine's constant |
2.68545200106530644... |
110,000 | X. Gourdon - 1998 |
G(1/4) |
3.62560990822190831... |
10,000,000,000 | S. Kondo & S. Pagliarulo - 2010 |
G(1/3) |
2.67893853470774763... |
10,000,000,000 | S. Kondo & S. Pagliarulo - 2009 |
Euler's constant g | 0.57721566490153286... |
29,844,489,545 | R. Chan & A.J. Yee - 2009 |
Catalan's constant G | 0.91596559417721901... |
31,026,000,000 | R. Chan & A.J. Yee - 2009 |
z(3) | 1.20205690315959428... |
31,026,000,000 | R. Chan & A.J. Yee - 2009 |
log 2 | 0.69314718055994530... |
31,026,000,000 | R. Chan & A.J. Yee - 2009 |
Golden ratio f | 1.61803398874989484... | 1,000,000,000,000 | A.J. Yee - 2010 |
e | 2.71828182845904523... | 1,000,000,000,000 | S. Kondo & A.J. Yee - 2010 |
Ö2 | 1.41421356237309504... | 1,000,000,000,000 | S. Kondo & A.J. Yee - 2010 |
p | 3.14159265358979323... |
5,000,000,000,000 |
S. Kondo & A.J. Yee - 2010 |
The last constants of this table are fully detailed in our pages. The other one are less famous and are introduced in Steven Finch's encyclopedia: Mathematical Constants (Cambridge University Press). Here we only give of them a short description as well as some previous known high accuracy computations.
B = (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + ...
Brun showed in 1919 that this sum converges to a finite limit. In order to estimate this sum the only known algorithms are variant of classical sieve to compute all twin primes as far as possible. This constant is famous because T. Nicely found the famous Pentium bug during its computation... To achieve his estimation, he needed to find all twin primes up to about 3.10^{15 }! Note that the twin prime's constant is used in the estimation of this constant. The latest computation of this constant was achieved up to 10^{16} by P. Sebah in July 2002 and required a cluster of 16 computers working during two months.
Other computations:
- E.S. Selmer: 2 (1942)
- C.E. Fröberg: 3 (1961)
- D. Shanks & J.W. Wrench: 4 (1974)
- J. Bohman: 5 (1974)
- R.P. Brent: 6 (1975)
- P. Sebah: 8 (1999)
Other computations:
- E. Wirsing: 20
- P. Flajolet & B. Vallée: 35 (1995)
- J. Hershberger: 40 (1997)
- P. Sebah: 100 (2000)
- K. Briggs: 385 (2003)
Other computations:
- J.W. Wrench: 45 (both constants 1961)
- T.O. Silva: 500 (only Artin's constant)
- G. Niklasch: 1,000 (1999)
Other computations:
- H.P. Robinson: 36 (1979?)
- A. Fransén & S. Wrigge: 80 (1984)
- W. A. Johnson: 300
1/2 + 1/3 + 1/5 + 1/7 + ... + 1/p - log log p
as p tends to infinity. To achieve this you need to evaluate the Zeta function to a great accuracy for successive integer values.
Other computations:
- G. Niklash: 1,000 (1999)
- P. Sebah: 4,050 (2000)
N(x) is about K.x/(log x)^{1/2}^{ }
Other computations:
- S. Ramanujan: 3
- P. Flajolet: 1,024 (1996)
- V. Adamchik, J. Golden & W. Gosper: 5,000
- D. Hare: 10,000 (1996)
Other computations:
- S. Ramanujan: 4
- J. von Soldner: 9
- P. Sebah: 10,000 (1999)
Other computations:
- R.W. Gosper: 2,217 (1997?)
- D.H. Bailey, J.M. Borwein & R.E. Crandall: 7,350 (1997)
Other computations:
- G.J. Fee & S. Plouffe: 250,000 (both constants in 1996)
- S. Spännare: 16,693,288 (2003)
- P. Sebah & M. Tommila: 51,097,000 (only G(1/4) in 2001)
G = 1/1^{2 }- 1/3^{2 } + 1/5^{2 }- 1/7^{2 }+ 1/9^{2} - ...^{ }
Other computations:
- J.W.L. Glaisher: 20 (1877)
- J.W.L. Glaisher: 32 (1913)
- G.J. Fee: 20,000 (1990)
- G.J. Fee: 50,000 (1996)
- G.J. Fee & S. Plouffe: 100,000 (1996)
- T. Papanikolaou: 300,000 (1996)
- T. Papanikolaou: 1,500,000 (1996)
- P. Demichel: 3,379,957 (1997)
- X. Gourdon: 12,500,000 (1998)
- X. Gourdon & P. Sebah: 50,010,000 (2001)
- X. Gourdon & P. Sebah: 100,000,500 (2001)
- S. Kondo & S. Pagliarulo: 2,000,000,000 (2007)
- S. Kondo & S. Pagliarulo: 10,000,000,000 (2008)
- R. Chan & A.J. Yee: 15,510,000,000 (2009)
Other computations:
- M. Maestlin: 7 (1597)
- M. Berg: 4,599 (1966)
- J. Shallit: 10,000 (1976)
- G.J. Fee & S. Plouffe: 10,000,000 (1996)
- X. Gourdon & P. Sebah: 1,500,000,000 (2000)
- X. Gourdon & P. Sebah: 3,141,000,000 (2002)
- A. Irlande: 5,000,000,000 (2007)
- S. Kondo & S. Pagliarulo: 10,000,000,000 (2007)
- A. Irlande: 10,000,000,000 (2007)
- X. Gourdon & P. Sebah: 16,380,340,000 (2008)
- A. Irlande: 17,000,000,000 (2008)
- X. Gourdon & P. Sebah: 31,415,927,000 (2008)
Please, let us know if the number of correct digits of any of those constants has been improved or if a previous computation should appear here.