Mathematical constant

This article describes special numbers that have special meaning in mathematics. See constant (logic) and constant function for other meanings of “constant” in mathematics.

A mathematical constant is a well-defined , real , non- integer number that is of particular interest in mathematics . Unlike physical constants , mathematical constants are defined independently of any physical measure. Many special numbers have special meanings in mathematics and appear in many different contexts. For example, there is exactly one differentiable function with and on the real or complex numbers . Consequently, a mathematical constant: . On the complex numbers is a periodic function , and its period length is another mathematical constant: . In many cases, mathematical constants can be calculated numerically with any precision. However, there are also some mathematical constants for which only very rough approximations are known, such as Brun's constant ${\displaystyle f}$ ${\displaystyle f^{\prime }=f}$ ${\displaystyle f(0)=1}$ ${\displaystyle f(1)}$ ${\displaystyle e}$ ${\displaystyle f}$ ${\displaystyle 2\pi }$ ${\displaystyle B_{2}=1{,}90216058\dots }$

Mathematical constants are examined in different sub-areas of mathematics. Despite great efforts, most mathematical constants are unclear whether they are rational , irrational - algebraic or transcendent . The polylogarithmic constants , to which the logarithms and the values ​​of the Riemann zeta function at the positive integer argument positions belong, form a particularly simple class . BBP series are known for part of this class .

Some important math constants

symbol	Decimal ( OEIS link)	Name and formula	Number type	First described	Number of known decimal places	description
${\displaystyle \pi }$	= 3.14159 26535 89793 23846… ( A000796 )	Kreiszahl , Pi, Archimedes constant ludolphsche number	transcendently predictable	2000 BC Chr.	50 · 10 12	Ratio of the circumference to the diameter of a circle .
${\displaystyle {\sqrt {2}}}$	= 1.41421 35623 73095 04880… ( A002193 )	Square root of 2 , Pythagorean constant	irrational algebraic	800 BC Chr.	10 13	Ratio of the diagonals to the edge length of a square ; positive solution of ${\displaystyle x^{2}=2}$
${\displaystyle {\sqrt {3}}}$	= 1.73205 08075 68877 29352… ( A002194 )	Square root of 3 , Theodorus constant	irrational algebraic	800 BC Chr.	2 · 10 12	Ratio of the spatial diagonals to the edge length of a cube ; positive solution of ${\displaystyle x^{2}=3}$
${\displaystyle \varphi ,\tau }$	= 1.61803 39887 49894 84820 ... ( A001622 )	Golden ratio : ${\displaystyle \textstyle {\frac {1+{\sqrt {5}}}{2}}}$	irrational algebraic	250 BC Chr.	6 · 10 12	Size ratio that often appears approximately in animate and inanimate nature - particularly irrational in a mathematically precise sense; positive solution of ${\displaystyle x^{2}=x+1}$
${\displaystyle \mathrm {e} }$	= 2.71828 18284 59045 23536 ... ( A001113 )	Euler's number : ${\displaystyle \textstyle \sum \limits _{k=0}^{\infty }{\frac {1}{k!}}}$	transcendently predictable	1618 1683	12 · 10 12	Base of the natural logarithm
${\displaystyle \gamma }$	= 0.57721 56649 01532 86060… ( A001620 )	Euler-Mascheroni constant : ${\displaystyle \textstyle \lim \limits _{n\to \infty }{\Bigl (}\sum \limits _{k=1}^{n}\!{\frac {1}{k}}-\ln n{\Bigr )}}$	predictable	1734	6 · 10 11	Area between the hyperbola and the stairs for ${\displaystyle {\frac {1}{x}}}$ ${\displaystyle {\frac {1}{\lfloor x\rfloor }}}$ ${\displaystyle x\geq 1}$
${\displaystyle \zeta (3)}$	= 1.20205 69031 59594 28539… ( A002117 )	Apéry constant : ${\displaystyle \textstyle \sum \limits _{k=1}^{\infty }{\frac {1}{k^{3}}}}$	irrationally predictable	1735	12 · 10 11	Value of the Riemann zeta function at point 3; The reciprocal of the asymptotic probability that 3 randomly chosen natural numbers are coprime ${\displaystyle \zeta (3)}$
${\displaystyle E_{B}}$	= 1.60669 51524 15291 76378… ( A065442 )	Erdős-Borwein constant : ${\displaystyle \textstyle \sum \limits _{n=1}^{\infty }{\frac {1}{2^{n}-1}}}$	irrational	1749	2000 ( OEIS )	Sum of the reciprocal values ​​of all Mersenne numbers
${\displaystyle \mu }$	= 1.45136 92348 83381 05028… ( A070769 )	Ramanujan-Soldner constant		1792 1809	75,500	Zero of the integral logarithm
${\displaystyle \varpi }$	= 2.62205 75542 92119 81046… ( A062539 )	Lemniscatic constant : ${\displaystyle \textstyle 2\int _{0}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{4}}}}}$	transcendently predictable	1798	6 · 10 11	Analog of π for the lemniscate
${\displaystyle B_{L}}$	= 1.08366.	Legendre constant	rational	1808	(5)	from Legendre's estimate x  / ( ln  x  - 1.08366) of the number of prime numbers ≤  x ; 1 is asymptotically correct
	= 0.66274 34193 49181 58097… ( A033259 )	Limit of Laplace		1827	500	maximum eccentricity for which the Laplace series converges to solve the Kepler equation
${\displaystyle G}$	= 0.91596 55941 77219 01505 ... ( A006752 )	Catalan's constant : ${\displaystyle \textstyle \sum \limits _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}}$	predictable	1832 1864	6 · 10 11	Value β (2) of the Dirichlet beta function at point 2
M 1	= 0.26149 72128 47642 78375 ... ( A077761 )	Meissel-Mertens constant : ${\displaystyle \textstyle \lim \limits _{n\to \infty }{\Bigl (}\!\sum \limits _{p\leq n \atop p\;{\text{prim}}}\!\!{\frac {1}{p}}-\ln \ln n{\Bigr )}}$		1866 1873	8010	Prime analogue of the Euler-Mascheroni constant
A.	= 1.28242 71291 00622 63687… ( A074962 )	Glaisher-Kinkelin constant : ${\displaystyle \textstyle \exp({\frac {1}{12}}-\zeta '(-1))}$		1856 1878	20,000	occurs in the evaluation of integrals and series sums on
C.	= 0.64341 05462 88338 02618 ... ( A118227 )	Cahen's constant : with , ${\displaystyle \sum \limits _{k=0}^{\infty }{\frac {(-1)^{k}}{S_{k}-1}}}$ ${\displaystyle S_{0}=2}$ ${\displaystyle S_{n}=1+S_{0}\cdots S_{n-1}}$	transcendently predictable	1891	4000	transcendental number with a simple formation law for the denominators of the continued fraction expansion
K	= 2.58498 17595 79253 21706… ( A062089 )	Sierpiński constant : ${\displaystyle \pi (2\gamma +4\ln \Gamma ({\tfrac {3}{4}})-\ln \pi )}$		1907	5000 ( OEIS )	occurs in the estimation of sums over τ ( n ) ƒ ( n on), where τ ( n ) is the number of the pairs ( a , b ) of integers with a 2 + b 2 =  n is
K	= 0.76422 36535 89220 66299 ... ( A064533 )	Landau-Ramanujan constant : ${\displaystyle \textstyle {\frac {1}{\sqrt {2}}}\!\!\!\!\prod \limits _{p\;{\text{prim}} \atop \equiv 3\;({\text{mod}}\;4)}\!\!\!\!{\bigl (}1-{\frac {1}{p^{2}}}{\bigr )}^{-1/2}}$		1908	125.079 ( OEIS )	the number of numbers ≤ x that are the sum of two square numbers  is ~  K  x / √ ln ( x )
G	= 1.01494 16064 09653 62502… ( A143298 )	Gieseking constant : ${\displaystyle \textstyle \int _{0}^{2\pi /3}\ln(2\cos(x/2))\,\mathrm {d} x}$		1912	105 (OEIS)	maximum volume of a hyperbolic tetrahedron
β	= 0.28016 94990 23869 13303 ... ( A073001 )	Bernstein constant		1913	50 (OEIS)	is the error of the best uniform approximation of | x | on [−1,1] by polynomials of even degree  n is ~  β / n
B 2	= 1.90216 058 ... ( A065421 )	Brun's constant : ${\displaystyle \textstyle \sum \limits _{p,\,p+2\;{\text{prim}}}\!{\bigl (}{\frac {1}{p}}+{\frac {1}{p+2}}{\bigr )}}$		1919	9 under the Hardy-Littlewood conjecture and a.	Sum of the reciprocal values ​​of all prime twins
Π 2 , C 2	= 0.66016 18158 46869 57392… ( A005597 )	Prime number twin constant : ${\displaystyle \textstyle \prod \limits _{p>2 \atop p\;{\text{prim}}}\!\!{\bigl (}1\!-\!{\frac {1}{(p-1)^{2}}}{\bigr )}}$		1922	5020	the number of prime twins ≤  x according to the Hardy-Littlewood conjecture ${\displaystyle \textstyle \sim 2\,C_{2}\int _{2}^{x}\!{\frac {\mathrm {d} t}{(\ln t)^{2}}}}$
?	> 0.5 + 10 −335 ≤ 0.54325 89653 42976 70695… ( A081760 )	Landau's constant		1929	1	Maximum, so that for every holomorphic function ƒ with ƒ ′ (0) = 1 in the image of the unit disk there is a circular disk with radius
λ, μ	= 0.62432 99885 43550 87099… ( A084945 )	Golomb-Dickman constant : ${\displaystyle \textstyle \int _{0}^{1}e^{{\rm {li}}(x)}\mathrm {d} x}$		1930 1964	1659	asymptotic mean relative length of the longest cycle of a permutation
K 0	= 2.68545 20010 65306 44530 ... ( A002210 )	Chinchin constant : ${\displaystyle \textstyle \prod \limits _{n=1}^{\infty }{\bigl (}1+{\frac {1}{n(n+2)}}{\bigr )}^{\log _{2}n}}$		1934	110,000	almost everywhere the geometric mean of the denominators of the continued fraction expansion
m	= 1.18656 91104 15625 45282… ( A100199 )	Chinchin-Lévy constant : ${\displaystyle \pi ^{2}/(12\,\ln 2)}$		1935	3.1026 · 10 10	almost everywhere the limit for n → ∞ of ( ln  q n ) / n , where q n of the denominator of the n -th proximity breakage is
A, θ	= 1.30637 78838 63080 69046… ( A051021 )	Mills constant		1946	6850 under the Riemann hypothesis	smallest number A > 0, such that ⌊ A 3 ⁿ ⌋ for each n = 1, 2, 3, ..., a prime number is
Λ	> −2.7 × 10 −9 <0.5	De Bruijn Newman constant		1948 1976	0	Minimum, so that a certain complex function H Λ has only real zeros; “Λ ≤ 0” is equivalent to the Riemann hypothesis
W.	= 1.53960 07178 39002 03869… ( A118273 )	Liebs ice cube constant : ${\displaystyle (4/3)^{3/2}}$	irrational algebraic	1967	1.6 · 10 8	Residual entropy of ice is N  k  ln  W in an exactly solvable 2D model in statistical physics
	= 1.70521 11401 05367 76428 ... ( A033150 )	Niven constant : ${\displaystyle \textstyle 1+\sum \limits _{k=2}^{\infty }{\bigl (}1-{\frac {1}{\zeta (k)}}{\bigr )}}$		1968	256	mean maximum exponent of the prime factorization of the numbers 1, 2, 3, ...
λ	= 0.30366 30028 98732 65859… ( A038517 )	Gauss-Kusmin-Savoy constant		1973	468	which comes in the description of convergence number distribution in continued fraction expansions on
C.	= 1.46707 80794 33975 47289 ... ( A086237 )	Porter's constant : ${\displaystyle \textstyle {\frac {6\ln 2}{\pi ^{2}}}{\bigl (}3\ln 2+4\gamma -{\frac {24}{\pi ^{2}}}\zeta '(2)-2{\bigr )}-{\frac {1}{2}}}$		1974	256	occurs in the formulas of the asymptotic mean division number in the Euclidean algorithm to
Ω	≈ 0.00787 49969 97812 3844 ( A100264 )	Chaitin's constant	unpredictable	1975	(64 bit)	Probability with which a universal Turing machine will stop on any input
α	= 0.80939 40205 40639 13071… ( A085291 )	Alladi-Grinstead constant : ${\displaystyle \textstyle \exp {\Bigl (}\!{\Bigl (}\sum \limits _{k=2}^{\infty }{\frac {1}{k}}\ln {\frac {k}{k-1}}{\Bigr )}-1{\Bigr )}}$		1977	102 (OEIS)	in n ! as the product of n prime powers, the largest possible smallest factor grows logarithmically ~  α  ln  n
δ	= 4.66920 16091 02990 67185… ( A006890 )	1. Fig tree constant		1979	1019	Transition into chaos : speed of bifurcation
α	= 2.50290 78750 95892 82228… ( A006891 )	2. Fig tree constant		1979	1019	Transition into chaos : reduction parameters
F.	= 2.80777 02420 28519 36522 ... ( A058655 )	Fransén-Robinson constant : ${\displaystyle \textstyle \int _{0}^{\infty }\!{\frac {1}{\Gamma (x)}}\,\mathrm {d} x}$		1978	1025	Area between the x -axis and the curve 1 / Γ ( x ) for x  > 0
Λ	= 1.09868 58055 25187 01 ... ( A086053 )	Lengyel's constant		1984	18 (OEIS)	occurs at the asymptotic analysis of the number of the chains from the smallest to the largest element in the association of the partitions on
σ	= 0.35323 63718 54995 98454 ... ( A085849 )	Hafner-Sarnak-McCurley constant : ${\displaystyle \textstyle \prod \limits _{p\;{\text{prim}}}\!\!\!{\Bigl (}\!1\!-\!{\bigl (}1\!-\!\!\prod \limits _{k=1}^{\infty }(1\!-\!{\frac {1}{p^{k}}}\!){\bigr )}^{\!2}\!{\Bigr )}}$		1993	40 (OEIS)	asymptotic probability that the determinants of two integer matrices are relatively prime
B.	= 1.45607 49485 82689 67139… ( A072508 )	Backhouse constant		1995	1300	−1 / B is the zero of the power series with 1 and the prime numbers as coefficients
K	= 1.13198 82487 943 ... ( A078416 )	Viswanath constant		1997	13 (OEIS)	Basis of the asymptotically exponential growth of random Fibonacci sequences
β *	= 0.70258 ... ( A118288 )	Embree-Trefethen constant		1999	5 (OEIS)	Boundary coefficient of generalized random Fibonacci sequences

See also

• List of special numbers
• Smarandache constants
• List of math symbols

literature

• Steven R. Finch: Mathematical constants , Cambridge University Press, Cambridge 2003, ISBN 0-521-81805-2 (English; Finch's website for the book with errata and addenda: Mathematical Constants . )

Web links

• Eric W. Weisstein : Constants . In: MathWorld (English).
• Simon Plouffe : Plouffe's inverter . (English) Enter a numerical value and search for the constant
• Anne Veling: Earliest Uses of Symbols for Constants . ( Memento of October 14, 2007 in the Internet Archive ) (Origin of the symbols) August 22, 1997 (English)
• Xavier Gourdon, Pascal Sebah: Numbers, constants and computation . (English)
• Alexander J. Yee: numberworld.org (English)
• Shigeru Kondo: PI World of JA0HXV . ( Memento from August 18, 2011 in the Internet Archive )

Individual evidence

1. ↑ Eric W. Weisstein : Constant . In: MathWorld (English).
2. ↑ Ferdinand von Lindemann : About the number π . (April and June 1882). In: Mathematische Annalen , 20, 1882, pp. 213–225, Textarchiv - Internet Archive
3. ^ Timothy Mullican: Calculating Pi: My attempt at breaking the Pi World Record. June 26, 2019, accessed March 14, 2020 (American English). 
4. ↑ ^{a b c d e f g h} Alexander J. Yee: Records Set by y-cruncher . numberworld.org, August 9, 2019 (English)
5. ^ Square Root of 2 . At: numberworld.org. January 9, 2017, accessed April 24, 2018.
6. ↑ Charles Hermite : Sur la fonction exponential . In: Comptes rendus des séances de l'Académie des sciences , 77, 1873, pp. 18–24 74–79 226–233 285–293 (French)
7. ↑ Jakob I Bernoulli , 1683, according to John J. O'Connor, Edmund F. Robertson : The number e . September 2001 (English)
8. ^ Leonhard Euler : De progressionibus harmonicis observationes . (March 11, 1734) In: Commentarii academiae scientiarum imperialis Petropolitanae , 7, 1740, pp. 150–161 (Latin; "C = 0.577218" on p. 157)
9. ↑ Roger Apéry : Irrationalité de ζ (2) et ζ (3) . In: Astérisque , 61, 1979, pp. 11-13 (French)
10. ↑ Leonhard Euler: Inventio summae cuiusque seriei ex dato termino generali . (October 13, 1735). In: Commentarii academiae scientiarum Petropolitanae , 8, 1741, pp. 9–22 (Latin; "1.202056903159594" on p. 21)
11. ^ Paul Erdős : On arithmetical properties of Lambert series (July 8, 1948). In : The Journal of the Indian Mathematical Society , 12, 1948, pp. 63–66 (English)
12. ↑ Leonhard Euler: Consideratio quarumdam serierum, quae singularibus proprietatibus sunt praeditae . (June 19, 1749 / January 26, 1750) In: Novi commentarii academiae scientiarum Petropolitanae , 3, 1753, pp. 86-108 (Latin; " s = 1.606695152415291" on p. 108)
13. ↑ Lorenzo Mascheroni : Adnotationes ad calculum integralem Euleri / In quibus nonnulla Problemata from Eulero proposita resolvuntur / Pars altera . Petrus Galeatius, Ticini 1792 (Latin; “z = 1.45137” on p. 17) Textarchiv - Internet Archive
14. ^ Johann Georg Soldner : Théorie et tables d'une nouvelle fonction transcendante . Lindauer, Munich 1809, p. 42 (French) Textarchiv - Internet Archive
15. ↑ ^{a b c d e f g} Xavier Gourdon, Pascal Sebah: Constants and Records of Computation . August 12, 2010
16. ^ Theodor Schneider : Arithmetic investigations of elliptical integrals . (March 11, 1936). In: Mathematische Annalen , 113, 1937, pp. 1-13
17. ^ Carl Friedrich Gauß , 1798
18. ^ Adrien-Marie Legendre : Essai sur la théorie des nombres . Duprat, Paris 1798, p. 19, Text Archive - Internet Archive . 2nd edition, Courcier, Paris 1808, p. 394 (French)
19. ↑ Pierre-Simon Laplace : Traité de mécanique céleste (Volume 5, Appendix). Bachelier, Paris 1827, p. 479 (French) Text archive - Internet Archive
    Félix Tisserand : Traité de mécanique céleste (Volume 1). Gauthier-Villars, Paris 1889, p. 262 (French) Textarchiv - Internet Archive
20. ^ The Laplace limit constant . ( Memento from March 17, 2011 in the Internet Archive ) at Plouffe's Inverter (English)
21. ^ Th. Clausen : About the Function ${\displaystyle \textstyle \sin \varphi +{\frac {1}{2^{2}}}\sin 2\varphi +{\frac {1}{3^{2}}}\sin 3\varphi +{\text{etc.}}}$ (March 3, 1832). In: Journal for pure and applied mathematics , 8, 1832, pp. 298-300 (“0.91596 55941 772190” on p. 300) Text archive - Internet Archive
22. ^ E. Catalan : Mémoire sur la transformation des séries, et sur quelques intégrales définies . In: Comptes rendus hebdomadaires des séances de l'Académie des sciences , 59, 1864, pp. 618–620 (French; “0.915 965 594 177 21” on p. 620) Text archive - Internet Archive
23. ↑ Ernst Meissel , Note, Nachr. Provinzial-Gewerbeschule Iserlohn, 1866 (in the estate)
24. ↑ Franz Mertens : A contribution to analytical number theory . (July 20, 1873). In: Journal for pure and applied mathematics , 78, 1874, pp. 46–62
25. ↑ Hermann Kinkelin : About a transcendent related to the gamma function and its application to the integral calculus . (July 1856). In: Journal for pure and applied mathematics , 57, 1860, pp. 122-138
26. ↑ JWL Glaisher : On the Product 1¹.2².3³ ... nⁿ . In: The Messenger of Mathematics , 7, 1878, pp. 43–47 (English; “ A = 1 · 28242 7130” on p. 43) Textarchiv - Internet Archive
27. ↑ 20,000 digits of the Glaisher-Kinkelin constant . ( Memento of the original from March 13, 2011 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. at the mpmath project (English) 
28. ↑ J. Les Davison, Jeffrey Shallit : Continued fractions for some alternating series (October 17, 1990). In: Monthly books for mathematics , 111, 1991, pp. 119–126 (English)
29. ↑ Eugène Cahen: Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues . In: Nouvelles Annales de Mathématiques , 10, 1891, pp. 508-514 (French) Text Archive - Internet Archive
30. ↑ The Cahen constant to 4000 digits . ( Memento from March 17, 2011 in the Internet Archive ) at Plouffe's Inverter (English)
31. ↑ Wacław Sierpiński : O sumowaniu szeregu , gdzie τ (n) oznacza liczbę rozkładów liczby n na sumę kwadratów dwóch liczb całkowitych ${\displaystyle \textstyle \sum _{n>a}^{n\leq b}\tau (n)f(n)}$ (About the summation of the series where τ ( n ) denotes the number of representations of n as the sum of two squares). In: Prace matematyczno-fizyczne , 18, 1907, pp. 1–59 (Polish; “ K = 2.5849817596” on p. 27) Textarchiv - Internet Archive ${\displaystyle \textstyle \sum _{n>a}^{n\leq b}\tau (n)f(n)}$
32. ↑ Edmund Landau : On the division of the positive whole numbers into four classes according to the minimum number of squares required for their additive composition . (June 21, 1908). In: Archive of Mathematics and Physics , 13, 1908, pp. 305-312, Textarchiv - Internet Archive
33. ↑ Hugo Gieseking : Analytical investigations on topological groups . L. Wiegand, Hilchenbach 1912 (inaugural dissertation at the Westphalian Wilhelms University of Münster)
34. ^ John W. Milnor : Hyperbolic geometry: The first 150 years . Bulletin of the AMS 6, 1982, pp. 9-24 (English)
35. ^ Serge Bernstein : Sur la meilleure approximation de | x | par des polynomes de degrés donnés . ( Memento of the original from January 30, 2012 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 2.2 MB; April 1913), Acta Mathematica 37, 1914, pp. 1–57 (French) 
36. ^ Viggo Brun : La serie ou les denominateurs sont "nombres premiers jumeaux" est convergente ou finie ${\displaystyle \textstyle {\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}+{\frac {1}{19}}+{\frac {1}{29}}+{\frac {1}{31}}+{\frac {1}{41}}+{\frac {1}{43}}+{\frac {1}{59}}+{\frac {1}{61}}+\ldots }$ . In: Bulletin des Sciences Mathématiques , 43, 1919, pp. 100–104 124–128 (French)
37. ^ GH Hardy , JE Littlewood : Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes . ( Memento of the original from January 30, 2012 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 2.5 MB; February 1922). In: Acta Mathematica , 44, 1923, pp. 1–70 (English) 
38. ↑ Edmund Landau : About the Bloch constant and two related world constants . (March 22, 1929). In: Mathematische Zeitschrift , December 30, 1929, pp. 608–634 (“ ” on p. 611, “ ” on p. 614) ${\displaystyle \textstyle {\mathfrak {L}}<{\frac {9}{16}}}$ ${\displaystyle {\mathfrak {L}}\;\geq \;0{,}43}$
39. ^ Lars Ahlfors : An extension of Schwarz's lemma . (April 1, 1937). In: Transactions of the AMS , 43, May 1938, pp. 359–364 (English; " L ≥1 / 2" on p. 364)
40. ^ Karl Dickman: On the frequency of numbers containing prime factors of a certain relative magnitude . In: Arkiv för Matematik, Astronomi och Fysik , 22A, 1930, pp. 1–14 (English)
41. ^ Solomon W. Golomb : Random permutations . (June 8, 1964). In: Bulletin of the AMS , 70, 1964, p. 747 (English; "λ = .62432965")
42. ^ David John Broadhurst: Titanic Golomb-Dickman prime . April 2, 2010
43. ↑ A. Khintchine : Metric continued fraction problems . (March 29, 1934). In: Compositio Mathematica , 1, 1935, pp. 361–382 (“2,6…” on p. 376)
44. ^ Paul Lévy : Sur le développement en fraction continue d'un nombre choisi au hasard . (July 1935). In: Compositio Mathematica , 3, 1936, pp. 286–303 (French)
45. ^ Xavier Gourdon, Pascal Sebah: Constants and Records of Computation . March 23, 2010 (English; π and ln 2 were calculated)
46. ^ William H. Mills: A prime-representing function . (December 23, 1946). In: Bulletin of the AMS , 53, 1947, p. 604 (English)
47. ↑ Chris K. Caldwell, Yuanyou Cheng: Determining Mills' Constant and a Note on Honaker's Problem . (August 15, 2005). In: Journal of Integer Sequences , 8, 2005, No. 05.4.1 (English)
48. ^ NG de Bruijn : The roots of trigonometric integrals . (PDF; 1.4 MB; July 16, 1948). In: Duke Mathematical Journal , September 17, 1950, pp. 197–226 (English)
49. ^ Charles M. Newman: Fourier transforms with only real zeros . (January / May 1976). In: Proceedings of the AMS , 61, December 1976, pp. 245-251 (English)
50. ^ Elliott H. Lieb : The residual entropy of square ice (May 22, 1967). In: Physical Review , 162, October 1967, pp. 162-172 (English)
51. ↑ Alexander J. Yee: Mathematical Constants - Millions of Digits . numberworld.org (English; was calculated √3 = ⁹⁄₈ W )
52. ↑ Ivan Niven : Averages of exponents in factoring integers . (June 18, 1968). In: Proceedings of the AMS , 22, 1969, pp. 356-360 (English)
53. ↑ The Niven constant is 1 + Sum (1-1 / Zeta (n), n = 2..infinity) . ( Memento from March 17, 2011 in the Internet Archive ) at Plouffe's Inverter (English)
54. ^ Eduard Wirsing: On the theorem of Gauss-Kusmin-Lévy and a Frobenius-type theorem for function spaces . (PDF; 796 kB; January 31, 1973). In: Acta Arithmetica , 24, 1974, pp. 507-528 (English; "λ = 0.3036630029 ..." on p. 509)
55. ^ JW Porter: On a theorem of Heilbronn (December 20, 1974). In: Mathematika , June 22, 1975, pp. 20–28 (English)
56. ^ The Porter constant . ( Memento from March 17, 2011 in the Internet Archive ) at Plouffe's Inverter (English)
57. ^ Gregory Chaitin : A theory of program size formally identical to information theory . (PDF; 249 kB; April / December 1974). In: Journal of the ACM , July 22, 1975, pp. 329-340 (English)
58. ↑ Krishnaswami Alladi, Charles Grinstead: On the decomposition of n! into prime powers . In: Journal of Number Theory , November 9, 1977, pp. 452–458 (English)
59. ^ ^{A b} Mitchell J. Feigenbaum : The universal metric properties of nonlinear transformations . (PDF; 1.39 MB; May 29, 1979). In: Journal of Statistical Physics , December 21, 1979, pp. 669–706 (English; "α = 2.502907876" on p. 703, "δ = 4.6692" on p. 704)
60. ↑ ^{a b} Feigenbaum constants to 1018 decimal places . ( Memento of March 17, 2011 in the Internet Archive ) at Plouffe's Inverter , March 22, 1999 (English)
61. ↑ Arne Fransén: Accurate determination of the inverse gamma integral (October 25, 1978). In: BIT Numerical Mathematics , March 19, 1979, pp. 137-138 (English)
62. ↑ Tamás Lengyel: On a recurrence involving Stirling numbers . In: European Journal of Combinatorics , 5, 1984, pp. 313–321 (English)
63. ↑ James Lee Hafner, Peter Sarnak , Kevin McCurley: Relatively prime values ​​of polynomials . (PDF; 174 kB) In: Marvin Knopp, Mark Sheingorn (Ed.): A tribute to Emil Grosswald : number theory and related analysis . In: AMS , Providence 1993, pp. 437-443 (English)
64. ↑ Simon Plouffe : The Backhouse constant calculated by Philippe Flajolet INRIA Paris to 1300 places . ( Memento of the original from February 5, 2008 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. August 1998 
65. ↑ Divakar Viswanath: Random Fibonacci sequences and the number 1.13198824… . (PDF; 484 kB) September 30, 1997 (English)
66. ↑ Mark Embree, Lloyd N. Trefethen: Growth and decay of random Fibonacci sequences . (PDF; 382 kB; September 18, 1998) In: Proceedings of the Royal Society A, 455, July 1999, pp. 2471–2485 (English)