User:Atavoidirc/constants

List of mathematical constants

A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. For example, the constant π may be defined as the ratio of the length of a circle's circumference to its diameter. The following list includes a decimal expansion and set containing each number, ordered by year of discovery.

The column headings may be clicked to sort the table alphabetically, by decimal value, or by set. Explanations of the symbols in the right hand column can be found by clicking on them.

List

Name	Symbol	Decimal expansion	Formula	Year	Set	Continued Fraction
One	1	1		Prehistory	$\mathbb {N}$	[1; ]
Two	2	2		Prehistory	$\mathbb {N}$	[2; ]
One half	1/2	0.5		Prehistory	$\mathbb {Q}$	[0; 2]
Pi	$\pi$	3.14159 26535 89793 23846	Ratio of a circle's circumference to its diameter.	1900 to 1600 BCE	$\mathbb {T}$	[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, …]
Square root of 2, Pythagoras constant.	${\sqrt {2}}$	1.41421 35623 73095 04880	Positive root of $x^{2}=2$	1800 to 1600 BCE	$\mathbb {A}$	[1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, …]
Square root of 3, Theodorus' constant	${\sqrt {3}}$	1.73205 08075 68877 29352	Positive root of $x^{2}=3$	465 to 398 BCE	$\mathbb {A}$
Square root of 5	${\sqrt {5}}$	2.23606 79774 99789 69640	Positive root of $x^{2}=5$		$\mathbb {A}$
Phi, Golden ratio	$\varphi$	1.61803 39887 49894 84820	${\frac {1+{\sqrt {5}}}{2}}$	~300 BCE	$\mathbb {A}$	[1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …]
Silver ratio	$\delta _{S}$	2.41421 35623 73095 04880	${\sqrt {2}}+1$	~300 BCE	$\mathbb {A}$
Zero	0	0		300 to 100 BCE	$\mathbb {Z}$	[0; ]
Negative one	−1	−1		300 to 200 BCE	$\mathbb {Z}$	[-1; ]
Cube root of 2 (Delian Constant)	${\sqrt[{3}]{2}}$	1.25992 10498 94873 16476	Real root of $x^{3}=2$	46 to 120 CE	$\mathbb {A}$
Cube root of 3	${\sqrt[{3}]{3}}$	1.44224 95703 07408 38232	Real root of $x^{3}=3$		$\mathbb {A}$
Twelfth root of 2	${\sqrt[{12}]{2}}$	1.05946 30943 59295 26456	Real root of $x^{12}=2$		$\mathbb {A}$
Supergolden ratio	$\psi$	1.46557 12318 76768 02665	${\frac {1+{\sqrt[{3}]{\frac {29+3{\sqrt {93}}}{2}}}+{\sqrt[{3}]{\frac {29-3{\sqrt {93}}}{2}}}}{3}}$ Real root of $x^{3}=x^{2}+1$		$\mathbb {A}$
Imaginary unit	$i$	0 + 1i	Either of the two roots of $x^{2}=-1$	1501 to 1576	$\mathbb {C}$	[i; ]
Wallis's constant		2.09455 14815 42326 59148	${\sqrt[{3}]{\frac {45-{\sqrt {1929}}}{18}}}+{\sqrt[{3}]{\frac {45+{\sqrt {1929}}}{18}}}$ Real root of $x^{3}-2x-5=0$	1616 to 1703	$\mathbb {A}$
Euler's number	$e$	2.71828 18284 59045 23536	$\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}=\sum _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}\cdots$	1618	$\mathbb {T}$	[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …]
Natural logarithm of 2	$\ln 2$	0.69314 71805 59945 30941	Real root of $e^{x}=2$ $\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+\cdots$	1619 & 1668	$\mathbb {T}$
Sophomore's dream1 J.Bernoulli	$I_{1}$	0.78343 05107 12134 40705	$\int _{0}^{1}\!x^{x}\,dx=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{n}}}={\frac {1}{1^{1}}}-{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}-\cdots$	1697
Sophomore's dream2 J.Bernoulli	$I_{2}$	1.29128 59970 62663 54040	$\int _{0}^{1}\!{\frac {dx}{x^{x}}}=\sum _{n=1}^{\infty }{\frac {1}{n^{n}}}={\frac {1}{1^{1}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{4}}}+\cdots$	1697
Lemniscate constant	$\varpi$	2.62205 75542 92119 81046	$\pi \,{G}=4{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {5}{4}}\right)^{2}}={\tfrac {1}{4}}{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {1}{4}}\right)^{2}}$ where $G$ is Gauss's constant	1718 to 1798	$\mathbb {T}$
Euler–Mascheroni constant	$\gamma$	0.57721 56649 01532 86060	$\lim _{n\to \infty }\left(-\log n+\sum _{k=1}^{n}{\frac {1}{k}}\right)=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,dx$	1735		[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …]
Erdős–Borwein constant	$E$	1.60669 51524 15291 76378	$\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}={\frac {1}{1}}\!+\!{\frac {1}{3}}\!+\!{\frac {1}{7}}\!+\!{\frac {1}{15}}\!+\!\cdots$	1749	$\mathbb {R} \setminus \mathbb {Q}$	[1; 1, 1, 1, 1, 5, 2, 1, 2, 29, 4, 1, 2, 2, 2, 2, 6, 1, 7, 1, 6, …]
Omega constant	$\Omega$	0.56714 32904 09783 87299	$W(1)={\frac {1}{\pi }}\int _{0}^{\pi }\log \left(1+{\frac {\sin t}{t}}e^{t\cot t}\right)dt$ where W is the Lambert W function	1758 & 1783	$\mathbb {T}$	[0; 1, 1, 3, 4, 2, 10, 4, 1, 1, 1, 1, 2, 7, 306, 1, 5, 1, 2, 1, 5, …]
Laplace limit		0.66274 34193 49181 58097	Real root of ${\frac {xe^{\sqrt {x^{2}+1}}}{{\sqrt {x^{2}+1}}+1}}=1$	~1782	$\mathbb {T}$
Gauss's constant	$G$	0.83462 68416 74073 18628	${\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}={\frac {\Gamma ({\frac {1}{4}})^{2}}{2{\sqrt {2\pi ^{3}}}}}={\frac {2}{\pi }}\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}$ where agm is the arithmetic–geometric mean	1799	$\mathbb {T}$	[0; 1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, …]
Ramanujan–Soldner constant	$\mu$	1.45136 92348 83381 05028	$\mathrm {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}=0$; root of the logarithmic integral function.	1812		[1; 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, 4, 1, 12, 1, 1, 2, 2, 1, …]
Second Hermite constant	$\gamma _{2}$	1.15470 05383 79251 52901	${\frac {2}{\sqrt {3}}}$	1822 to 1901	$\mathbb {A}$
Liouville's constant	$L$	0.11000 10000 00000 00000 0001	$\sum _{n=1}^{\infty }{\frac {1}{10^{n!}}}={\frac {1}{10^{1!}}}+{\frac {1}{10^{2!}}}+{\frac {1}{10^{3!}}}+{\frac {1}{10^{4!}}}+\cdots$	Before 1844	$\mathbb {T}$
First continued fraction constant	$C_{1}$	0.69777 46579 64007 98201	${\tfrac {1}{1+{\tfrac {1}{2+{\tfrac {1}{3+{\tfrac {1}{4+{\tfrac {1}{5+\cdots }}}}}}}}}}$ ${\frac {I_{1}(2)}{I_{0}(2)}}$, where $I_{\alpha }(x)$ is the modified Bessel function	1855	$\mathbb {R} \setminus \mathbb {Q}$	[0; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …]
Hermite–Ramanujan constant		262 53741 26407 68743 .99999 99999 99250 073	$e^{\pi {\sqrt {163}}}$	1859	$\mathbb {T}$
Glaisher–Kinkelin constant	$A$	1.28242 71291 00622 63687	$e^{{\frac {1}{12}}-\zeta ^{\prime }(-1)}=e^{{\frac {1}{8}}-{\frac {1}{2}}\sum \limits _{n=0}^{\infty }{\frac {1}{n+1}}\sum \limits _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}\left(k+1\right)^{2}\ln(k+1)}$	1860 to 1894
Catalan's constant	$G$	0.91596 55941 77219 01505	$\int _{0}^{1}\!\!\int _{0}^{1}\!\!{\frac {dx\,dy}{1{+}x^{2}y^{2}}}=\!\sum _{n=0}^{\infty }\!{\frac {(-1)^{n}}{(2n{+}1)^{2}}}\!=\!{\frac {1}{1^{2}}}{-}{\frac {1}{3^{2}}}{+}{\cdots }$	1864		[0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …]
Dottie number		0.73908 51332 15160 64165	Real root of $\cos x=x$	1865	$\mathbb {T}$
Meissel–Mertens constant	$M$	0.26149 72128 47642 78375	$\lim _{n\to \infty }\left(\sum _{p\leq n}{\frac {1}{p}}\ln \ln n\right)=\gamma +\sum _{p}\left(\ln \left(1-{\frac {1}{p}}\right)+{\frac {1}{p}}\right)$ where γ is the Euler–Mascheroni constant and p is prime	1866 & 1873		[0; 3, 1, 4, 1, 2, 5, 2, 1, 1, 1, 1, 13, 4, 2, 4, 2, 1, 33, 296, 2, …]
Weierstrass constant		0.47494 93799 87920 65033	${\frac {2^{5/4}{\sqrt {\pi }}\,e^{\pi /8}}{\Gamma ({\frac {1}{4}})^{2}}}$	1872 ?	$\mathbb {T}$
First Hafner–Sarnak–McCurley constant	$D(1)$	0.60792 71018 54026 62866	${\frac {6}{\pi ^{2}}}=\prod _{p{\text{ prime}}}{\left(1-{\frac {1}{p^{2}}}\right)}\!=\textstyle \left(1-{\frac {1}{2^{2}}}\right)\left(1-{\frac {1}{3^{2}}}\right)\cdots$	1883	$\mathbb {T}$
Universal parabolic constant	$P$	2.29558 71493 92638 07403	$\ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arsinh} (1)+{\sqrt {2}}$	Before 1891	$\mathbb {T}$	[2; 3, 2, 1, 1, 1, 1, 3, 3, 1, 1, 4, 2, 3, 2, 7, 1, 6, 1, 8, 7, …]
Cahen's constant	$C$	0.64341 05462 88338 02618	$\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{s_{k}-1}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{6}}-{\frac {1}{42}}+{\frac {1}{1806}}{\,\pm \cdots }$ where sk is the kth term of Sylvester's sequence 2, 3, 7, 43, 1807, ...	1891	$\mathbb {T}$	[0; 1, 1, 1, 22, 32, 132, 1292, 252982, 4209841472, 2694251407415154862, …]
Apéry's constant	$\zeta (3)$	1.20205 69031 59594 28539	$\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+\cdots$	1895	$\mathbb {R} \setminus \mathbb {Q}$	[1; 4, 1, 18, 1, 1, 1, 4, 1, 9, 9, 2, 1, 1, 1, 2, 7, 1, 1, 7, 11, …]
Gelfond's constant	$e^{\pi }$	23.14069 26327 79269 0057	$(-1)^{-i}=i^{-2i}=\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}=1+{\frac {\pi ^{1}}{1}}+{\frac {\pi ^{2}}{2}}+{\frac {\pi ^{3}}{6}}+\cdots$	1900	$\mathbb {T}$	[23; 7, 9, 3, 1, 1, 591, 2, 9, 1, 2, 34, 1, 16, 1, 30, 1, 1, 4, 1, 2, …]
Second Favard constant	$K_{2}$	1.23370 05501 36169 82735	${\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots$	1902 to 1965	$\mathbb {T}$
Golden angle	$g$	2.39996 32297 28653 32223	${\frac {2\pi }{\varphi ^{2}}}=\pi (3-{\sqrt {5}})$ or $180(3-{\sqrt {5}})=137.50776\ldots$ in degrees	1907	$\mathbb {T}$
Sierpiński's constant	$K$	2.58498 17595 79253 21706	${\begin{aligned}&\pi \left(2\gamma +\ln {\frac {4\pi ^{3}}{\Gamma ({\tfrac {1}{4}})^{4}}}\right)=\pi (2\gamma +4\ln \Gamma ({\tfrac {3}{4}})-\ln \pi )\\&=\pi \left(2\ln 2+3\ln \pi +2\gamma -4\ln \Gamma ({\tfrac {1}{4}})\right)\end{aligned}}$	1907
First Nielsen–Ramanujan constant	$a_{1}$	0.82246 70334 24113 21823	${\frac {{\zeta }(2)}{2}}={\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}\cdots$	1909	$\mathbb {T}$
Gieseking constant	$G$	1.01494 16064 09653 62502	${\frac {3{\sqrt {3}}}{4}}\left(1-\sum _{n=0}^{\infty }{\frac {1}{(3n+2)^{2}}}+\sum _{n=1}^{\infty }{\frac {1}{(3n+1)^{2}}}\right)=$ $\textstyle {\frac {3{\sqrt {3}}}{4}}\left(1-{\frac {1}{2^{2}}}+{\frac {1}{4^{2}}}-{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}-{\frac {1}{8^{2}}}+{\frac {1}{10^{2}}}\pm \cdots \right)$.	1912
Bernstein's constant	$\beta$	0.28016 94990 23869 13303	$\lim _{n\to \infty }2nE_{2n}(f)$, where En(f) is the error of the best uniform approximation to a real function f(x) on the interval [−1, 1] by real polynomials of no more than degree n, and f(x) = |x|	1913		[0; 3, 1, 1, 3, 9, 6, 3, 1, 3, 13, 1, 16, 3, 3, 4, …]
Tribonacci constant		1.83928 67552 14161 13255	${\textstyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}={\frac {1+4\cosh \left({\frac {1}{3}}\cosh ^{-1}\left(2+{\frac {3}{8}}\right)\right)}{3}}}$ Real root of $x^{3}-x^{2}-x-1=0$	1914 to 1963	$\mathbb {A}$
Twin primes constant	$C_{2}$	0.66016 18158 46869 57392	$\prod _{\textstyle {p\;{\rm {prime}} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)$	1922		[0; 1, 1, 1, 16, 2, 2, 2, 2, 1, 18, 2, 2, 11, 1, 1, 2, 4, 1, 16, 3, …]
Z score for the 97.5 percentile point	$z_{.975}$	1.95996 39845 40054 23552	${\sqrt {2}}\operatorname {erf} ^{-1}(0.95)$ where erf-1(x) is the inverse error function Real number $z$ such that ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-x^{2}/2}\,\mathrm {d} x=0.975$	1925
Bloch–Landau constant	$L$	0.54325 89653 42976 70695	${\frac {\Gamma ({\tfrac {1}{3}})\;\Gamma ({\tfrac {5}{6}})}{\Gamma ({\tfrac {1}{6}})}}$	1929	$\mathbb {T}$
Plastic number	$\rho$	1.32471 79572 44746 02596	${\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\cdots }}}}}}=\textstyle {\sqrt[{3}]{{\frac {1}{2}}+{\frac {\sqrt {69}}{18}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {\sqrt {69}}{18}}}}$ Real root of $x^{3}=x+1$	1929	$\mathbb {A}$	[1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80, 2, 5, 1, 2, 8, 2, 1, 1, …]
Golomb–Dickman constant	$\lambda$	0.62432 99885 43550 87099	$\int _{0}^{1}e^{\mathrm {Li} (t)}dt=\int _{0}^{\infty }{\frac {\rho (t)}{t+2}}dt$ where Li(t) is the logarithmic integral, and ρ(t) is the Dickman function	1930 & 1964
Feller–Tornier constant	${\mathcal {C}}_{\mathrm {FT} }$	0.66131 70494 69622 33528	${{\frac {1}{2}}\prod _{p{\text{ prime}}}\left(1-{\frac {2}{p^{2}}}\right)+{\frac {1}{2}}}={\frac {3}{\pi ^{2}}}\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p^{2}-1}}\right)+{\frac {1}{2}}$	1932
Base 10 Champernowne constant	$C_{10}$	0.12345 67891 01112 13141	Defined by concatenating representations of successive integers: 0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...	1933	$\mathbb {T}$	[0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 4.57540×10165, 6, 1, …]
Gelfond–Schneider constant	$2^{\sqrt {2}}$	2.66514 41426 90225 18865		1934	$\mathbb {T}$	[2; 1, 1, 1, 72, 3, 4, 1, 3, 2, 1, 1, 1, 14, 1, 2, 1, 1, 3, 1, 3, …]
Khinchin's constant	$K_{0}$	2.68545 20010 65306 44530	$\prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\log _{2}(n)}$	1934		[2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …]
Khinchin–Lévy constant (1)	$\beta$	1.18656 91104 15625 45282	${\frac {\pi ^{2}}{12\,\ln 2}}$	1935
Khinchin-Lévy constant (2)	$e^{\beta }$	3.27582 29187 21811 15978	$e^{\pi ^{2}/(12\ln 2)}$	1936
Mills' constant	$A$	1.30637 78838 63080 69046	Smallest positive real number A such that $\lfloor A^{3^{n}}\rfloor$ is prime for all positive integers n	1947
Euler–Gompertz constant	$\delta$	0.59634 73623 23194 07434	$\int _{0}^{\infty }\!\!{\frac {e^{-x}}{1+x}}\,dn=\!\!\int _{0}^{1}\!\!{\frac {dx}{1-\ln x}}={\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {2}{1+{\tfrac {2}{1+{\tfrac {3}{1+3{/\cdots }}}}}}}}}}}}}$	Before 1948
de Bruijn–Newman constant	$\Lambda$	$0\leq \Lambda \leq 0.2$	The number Λ where for where $H(\lambda ,z)=\int _{0}^{\infty }e^{\lambda u^{2}}\Phi (u)\cos(zu)du$ has real zeros if and only if λ ≥ Λ. where $\Phi (u)=\sum _{n=1}^{\infty }(2\pi ^{2}n^{4}e^{9u}-3\pi n^{2}e^{5u})e^{-\pi n^{2}e^{4u}}$.	1950
Van der Pauw constant	${\frac {\pi }{\ln 2}}$	4.53236 01418 27193 80962		Before 1958	$\mathbb {R} \setminus \mathbb {Q}$
Magic angle	$\theta _{\mathrm {m} }$	0.95531 66181 245092 78163	$\arctan {\sqrt {2}}=\arccos {\tfrac {1}{\sqrt {3}}}\approx \textstyle {54.7356}^{\circ }$	Before 1959	$\mathbb {T}$
Lochs constant	$L$	0.97027 01143 92033 92574	${\frac {6\ln 2\ln 10}{\pi ^{2}}}$	1964
Lieb's square ice constant		1.53960 07178 39002 03869	$\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}$	1967	$\mathbb {A}$
Baker constant		0.83564 88482 64721 05333	$\int _{0}^{1}{\frac {\mathrm {d} t}{1+t^{3}}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{3n+1}}={\frac {1}{3}}\left(\ln 2+{\frac {\pi }{\sqrt {3}}}\right)$	Before 1969	$\mathbb {T}$
Niven's constant	$C$	1.70521 11401 05367 76428	$1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)}}\right)$	1969
Porter's constant	$C$	1.46707 80794 33975 47289	${\frac {6\ln 2}{\pi ^{2}}}\left(3\ln 2+4\,\gamma -{\frac {24}{\pi ^{2}}}\,\zeta '(2)-2\right)-{\frac {1}{2}}$ where γ is the Euler–Mascheroni constant and ζ '(2) is the derivative of the Riemann zeta function evaluated at s = 2	1974
Feigenbaum constant δ	$\delta$	4.66920 16091 02990 67185	$\lim _{n\to \infty }{\frac {x_{n+1}-x_{n}}{x_{n+2}-x_{n+1}}}$ where the sequence xn is given by $x_{n+1}=ax_{n}(1-x_{n})$	1975		[4; 1, 2, 43, 2, 163, 2, 3, 1, 1, 2, 5, 1, 2, 3, 80, 2, 5, 2, 1, 1, …]
Chaitin's constants	$\Omega$	In general they are uncomputable numbers. But one such number is 0.00787 49969 97812 3844	$\sum _{p\in P}2^{-|p|}$ p: Halted program |p|: Size in bits of program p P: Domain of all programs that stop. See also: Halting problem	1975	$\mathbb {T}$
Fransén–Robinson constant	$F$	2.80777 02420 28519 36522	$\int _{0}^{\infty }{\frac {dx}{\Gamma (x)}}=e+\int _{0}^{\infty }{\frac {e^{-x}}{\pi ^{2}+\ln ^{2}x}}\,dx$	1978		[2; 1, 4, 4, 1, 18, 5, 1, 3, 4, 1, 5, 3, 6, 1, 1, 1, 5, 1, 1, 1, …]
Robbins constant	$\Delta (3)$	0.66170 71822 67176 23515	${\frac {4\!+\!17{\sqrt {2}}\!-6{\sqrt {3}}\!-7\pi }{105}}\!+\!{\frac {\ln(1\!+\!{\sqrt {2}})}{5}}\!+\!{\frac {2\ln(2\!+\!{\sqrt {3}})}{5}}$	1978	$\mathbb {T}$
Fractal dimension of the Cantor set	$\log _{3}2$	0.63092 97535 71457 43709	${\frac {\log 2}{\log 3}}$	Before 1979	$\mathbb {T}$
Feigenbaum constant α	$\alpha$	2.50290 78750 95892 82228	Ratio between the width of a tine and the width of one of its two subtines in a bifurcation diagram	1979		[2; 1, 1, 85, 2, 8, 1, 10, 16, 3, 8, 9, 2, 1, 40, 1, 2, 3, 2, 2, 1, …]
Connective constant for the hexagonal lattice	$\mu$	1.84775 90650 22573 51225	${\sqrt {2+{\sqrt {2}}}}$, as a root of the polynomial $x^{4}-4x^{2}+2=0$	1982	$\mathbb {A}$
Lehmer's conjecture constant	$\sigma _{10}$	1.17628 08182 59917 50654	Real root of $x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1=0$	1983?	$\mathbb {A}$
Chebyshev constant		0.59017 02995 08048 11302	${\frac {\Gamma ({\tfrac {1}{4}})^{2}}{4\pi ^{3/2}}}$	Before 1987	$\mathbb {T}$
Conway's constant	$\lambda$	1.30357 72690 34296 39125	Real root of the polynomial: ${\begin{smallmatrix}x^{71}-x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}-x^{60}\\-x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{50}\\+x^{49}+9x^{48}-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{40}\\-12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{30}\\-10x^{29}-3x^{28}+2x^{27}+9x^{26}-3x^{25}+14x^{24}-8x^{23}-7x^{21}+9x^{20}\\+3x^{19}\!-4x^{18}\!-10x^{17}\!-7x^{16}\!+12x^{15}\!+7x^{14}\!+2x^{13}\!-12x^{12}\!-4x^{11}\!-2x^{10}\\+5x^{9}+x^{7}-7x^{6}+7x^{5}-4x^{4}+12x^{3}-6x^{2}+3x-6\ =\ 0\quad \quad \quad \end{smallmatrix}}$	1987	$\mathbb {A}$
Prévost constant, Reciprocal Fibonacci constant	$\psi$	3.35988 56662 43177 55317	$\sum _{n=1}^{\infty }{\frac {1}{F_{n}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+\cdots$ where Fn is the nth Fibonacci number	Before 1988	$\mathbb {R} \setminus \mathbb {Q}$	[3; 2, 1, 3, 1, 1, 13, 2, 3, 3, 2, 1, 1, 6, 3, 2, 4, 362, 2, 4, 8, …]
Brun's constant	$B_{2}$	1.90216 05831 04	$\textstyle {\sum \limits _{p}({\frac {1}{p}}+{\frac {1}{p+2}})}=({\frac {1}{3}}\!+\!{\frac {1}{5}})+({\tfrac {1}{5}}\!+\!{\tfrac {1}{7}})+({\tfrac {1}{11}}\!+\!{\tfrac {1}{13}})+\cdots$ where the sum ranges over all primes p such that p + 2 is also a prime	1989		[1; 1, 9, 4, 1, 1, 8, 3, 4, 7, 1, 3, 3, 1, 2, 1, 1, 12, 4, 2, 1, …]
Hafner–Sarnak–McCurley constant	$\sigma$	0.35323 63718 54995 98454	$\prod _{k=1}^{\infty }\left\{1-\left[1-\prod _{j=1}^{n}\left(1-p_{k}^{-j}\right)\right]^{2}\right\}$ where pk is the kth prime number	1993
Fractal dimension of the Apollonian packing of circles		1.30568 6729 ≈ by Thomas & Dhar 1.30568 8 ≈ by McMullen		1994 to 1998
Backhouse's constant	$B$	1.45607 49485 82689 67139	$\lim _{k\to \infty }\left|{\frac {q_{k+1}}{q_{k}}}\right\vert \quad \scriptstyle {\text{where:}}\displaystyle \;\;Q(x)={\frac {1}{P(x)}}=\!\sum _{k=1}^{\infty }q_{k}x^{k}$ $P(x)=1+\sum _{k=1}^{\infty }{p_{k}x^{k}}=1+2x+3x^{2}+5x^{3}+\cdots$where pk is the kth prime number	1995		[1; 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, 13, 3, 1, 2, 4, 16, 4, …]
Viswanath constant		1.13198 82487 943	$\lim _{n\to \infty }|f_{n}|^{\frac {1}{n}}$      where fn = fn−1 ± fn−2, where the signs + or − are chosen at random with equal probability 1/2	1997		[1; 7, 1, 1, 2, 1, 3, 2, 1, 2, 1, 17, 1, 1, 2, 1, 2, 4, 1, 2, …]
Regular paperfolding sequence	$P$	0.85073 61882 01867 26036	$\sum _{n=0}^{\infty }{\frac {8^{2^{n}}}{2^{2^{n+2}}-1}}=\sum _{n=0}^{\infty }{\cfrac {\tfrac {1}{2^{2^{n}}}}{1-{\tfrac {1}{2^{2^{n+2}}}}}}$	Before 1998	$\mathbb {T}$
Komornik–Loreti constant	$q$	1.78723 16501 82965 93301	Real number $q$ such that $1=\sum _{n=1}^{\infty }{\frac {t_{k}}{q^{k}}}$, or $\prod _{n=0}^{\infty }\left(1-{\frac {1}{q^{2^{n}}}}\right)+{\frac {q-2}{q-1}}=0$ where tk is the kth term of the Thue–Morse sequence	1998	$\mathbb {T}$
Artin constant	$C_{\mathrm {Artin} }$	0.37395 58136 19202 28805	$\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p(p-1)}}\right)$	1999
Embree–Trefethen constant	$\beta ^{\star }$	0.70258	the number $\beta ^{\star }$ where the sequence $f_{n}=f_{n-1}\pm \beta f_{n-2}$ decays almost surely if $\beta >\beta ^{\star }$ and grows almost surely otherwise	1999		[0; 1, 2, 2, 1, 3, 5, 1, 2, 6, 1, 1, 5, …]
MRB constant	$S$	0.18785 96424 62067 12024	$\sum _{n=1}^{\infty }(-1)^{n}(n^{1/n}-1)=-{\sqrt[{1}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+\cdots$	1999		[0; 5, 3, 10, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 12, 2, 17, 2, 2, 1, 1, …]
Somos' quadratic recurrence constant	$\sigma$	1.66168 79496 33594 12129	$\prod _{n=1}^{\infty }n^{{1/2}^{n}}={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots$	1999
Foias constant α	$\alpha$	1.18745 23511 26501 05459	$x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}{\text{ for }}n=1,2,3,\ldots$ Foias constant is the unique real number such that if x1 = α then the sequence diverges to infinity	2000
Foias constant β	$\beta$	2.29316 62874 11861 03150	Real root of $x^{x+1}=(x+1)^{x}$	2000	$\mathbb {T}$
DeVicci's tesseract constant		1.00743 47568 84279 37609	The largest cube that can pass through in an 4D hypercube. Positive root of $4x^{8}{-}28x^{6}{-}7x^{4}{+}16x^{2}{+}16=0$	Before 2001	$\mathbb {A}$
Ramanujan nested radical		2.74723 82749 32304 33305	$\scriptstyle {\sqrt {5+{\sqrt {5+{\sqrt {5-{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5-\cdots }}}}}}}}}}}}}}\;=\textstyle {\frac {2+{\sqrt {5}}+{\sqrt {15-6{\sqrt {5}}}}}{2}}$	Before 2001	$\mathbb {A}$
Hausdorff dimension of the Sierpinski triangle	$\log _{2}3$	1.58496 25007 21156 18145	${\frac {\log 3}{\log 2}}$	Before 2002	$\mathbb {T}$
Heath-Brown–Moroz constant	$C$	0.00131 76411 54853 17810	$\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p}}\right)^{7}\left(1+{\frac {7p+1}{p^{2}}}\right)$	Before 2002
Lebesgue constant	$c$	0.98943 12738 31146 95174	$\lim _{n\to \infty }\!\!\left(\!{L_{n}{-}{\frac {4}{\pi ^{2}}}\ln(2n{+}1)}\!\!\right)\!{=}{\frac {4}{\pi ^{2}}}\!\left({\sum _{k=1}^{\infty }\!{\frac {2\ln k}{4k^{2}{-}1}}}{-}{\frac {\Gamma '({\tfrac {1}{2}})}{\Gamma ({\tfrac {1}{2}})}}\!\!\right)$	Before 2002
2nd du Bois-Reymond constant	$C_{2}$	0.19452 80494 65325 11361	${\frac {e^{2}-7}{2}}=\int _{0}^{\infty }\left|{{\frac {d}{dt}}\left({\frac {\sin t}{t}}\right)^{n}}\right|\,dt-1$	Before 2003	$\mathbb {T}$
Landau–Ramanujan constant	$K$	0.76422 36535 89220 66299	${\frac {1}{\sqrt {2}}}\prod _{{p\equiv 3{\text{ mod }}4} \atop p\;{\rm {prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{-{\frac {1}{2}}}}\!\!={\frac {\pi }{4}}\prod _{{p\equiv 1{\text{ mod }}4} \atop p\;{\rm {prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{\frac {1}{2}}}$	Before 2005		[0; 1, 3, 4, 6, 1, 15, 1, 2, 2, 3, 1, 23, 3, 1, 1, 3, 1, 1, 7, 2, …]
Stephens constant		0.57595 99688 92945 43964	$\prod _{p{\text{ prime}}}\left(1-{\frac {p}{p^{3}-1}}\right)$	Before 2005
Taniguchi constant		0.67823 44919 17391 97803	$\prod _{p{\text{ prime}}}\left(1-{\frac {3}{p^{3}}}+{\frac {2}{p^{4}}}+{\frac {1}{p^{5}}}-{\frac {1}{p^{6}}}\right)$	Before 2005
Prime constant	$\rho$	0.41468 25098 51111 66024	$\sum _{p{\text{ prime}}}{\frac {1}{2^{p}}}={\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{32}}+\cdots$	Before 2008	$\mathbb {R} \setminus \mathbb {Q}$
Raabe's formula		0.91893 85332 04672 74178	$\int _{0}^{1}\ln \Gamma (x)dx={\frac {1}{2}}\ln(2\pi )=-\zeta '(0)$	Before 2011
Copeland–Erdős constant	${\mathcal {C}}_{CE}$	0.23571 11317 19232 93137	Defined by concatenating representations of successive prime numbers: 0.2 3 5 7 11 13 17 19 23 29 31 37 ...	Before 2012	$\mathbb {R} \setminus \mathbb {Q}$
Kepler–Bouwkamp constant	$K'$	0.11494 20448 53296 20070	$\prod _{n=3}^{\infty }\cos \left({\frac {\pi }{n}}\right)=\cos \left({\frac {\pi }{3}}\right)\cos \left({\frac {\pi }{4}}\right)\cos \left({\frac {\pi }{5}}\right)...$	Before 2013
Prouhet–Thue–Morse constant	$\tau$	0.41245 40336 40107 59778	$\sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}$ where ${t_{n}}$ is the nth term of the Thue–Morse sequence	Before 2014	$\mathbb {T}$

See also

• Invariant (mathematics)
• List of mathematical symbols
• List of mathematical symbols by subject
• List of numbers
• List of physical constants
• Particular values of the Riemann zeta function
• Physical constant

Bibliography

• Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 2013-06-05. English translation by Catriona and David Lischka.
• Jensen, Johan Ludwig William Valdemar (1895), "Note numéro 245. Deuxième réponse. Remarques relatives aux réponses du MM. Franel et Kluyver", L'Intermédiaire des Mathématiciens, II: 346–347
• Cuyt, Annie A.M.; Petersen, Vigdis; Verdonk, Brigitte; Waadeland, Haakon; Jones, William B. (2008). "Mathematical constants". Handbook of Continued Fractions for Special Functions. Dordrecht, Netherlands: Springer Science + Business Media. ISBN 9781402069499.
• Borwein, Jonathan; van der Poorten, Alf; Shallit, Jeffrey; Zudilin, Wadim (2014). Neverending Fractions: An Introduction to Continued Fractions. Australian Mathematical Society Lecture Series. Vol. 23. Cambridge, United Kingdom: Cambridge University Press. ISBN 9780521186490. ISSN 0950-2815.

Further reading

• Wolfram, Stephen. "4: Systems Based on Numbers". Section 5: Mathematical Constants — Continued fractions. {{cite book}}: |work= ignored (help)

External links

• Inverse Symbolic Calculator, Plouffe's Inverter
• Constants - from Wolfram MathWorld
• On-Line Encyclopedia of Integer Sequences (OEIS)
• Steven Finch's page of mathematical constants
• Xavier Gourdon and Pascal Sebah's page of numbers, mathematical constants and algorithms

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