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.^[1] 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.

[1]

Name	Symbol	Decimal expansion	Formula	Year	Set
Name	Symbol	Decimal expansion	Formula	Year	$\mathbb {Q}$	$\mathbb {A}$	${\mathcal {P}}$	${\mathcal {E}}{\mathcal {P}}$
Zero	0	0	Additive identity of $\mathbb {C}$	300 to 100 BCE^[2]	✓	✓	✓	✓
One	1	1	Multiplicative identity of $\mathbb {C}$	Prehistory	✓	✓	✓	✓
Two	2	2		Prehistory	✓	✓	✓	✓
One half	⁠1/2⁠	0.5	Multiplicative inverse of 2	Prehistory	✓	✓	✓	✓
Pi	$\pi$	3.14159 26535 89793 23846 ^{[Mw 1]}^{[OEIS 1]}	Ratio of a circle's circumference to its diameter	1900 to 1600 BCE ^[3]	✗	✗	✓	✓
Tau	$\tau$	6.28318 53071 79586 47692^[4]^{[OEIS 2]}	Ratio of a circle's circumference to its radius; equal to $2\pi$	1900 to 1600 BCE ^[3]	✗	✗	✓	✓
Square root of 2, Pythagoras constant^[5]	${\sqrt {2}}$	1.41421 35623 73095 04880 ^{[Mw 2]}^{[OEIS 3]}	Positive root of $x^{2}=2$	1800 to 1600 BCE^[6]	✗	✓	✓	✓
Square root of 3, Theodorus' constant^[7]	${\sqrt {3}}$	1.73205 08075 68877 29352 ^{[Mw 3]}^{[OEIS 4]}	Positive root of $x^{2}=3$	465 to 398 BCE	✗	✓	✓	✓
Square root of 5^[8]	${\sqrt {5}}$	2.23606 79774 99789 69640 ^{[OEIS 5]}	Positive root of $x^{2}=5$		✗	✓	✓	✓
Phi, Golden ratio^[9]	$\varphi$ or $\phi$	1.61803 39887 49894 84820 ^{[Mw 4]}^{[OEIS 6]}	${\frac {1+{\sqrt {5}}}{2}}$	~300 BCE	✗	✓	✓	✓
Silver ratio^[10]	$\sigma$	2.41421 35623 73095 04880 ^{[Mw 5]}^{[OEIS 7]}	${\sqrt {2}}+1$	~300 BCE	✗	✓	✓	✓
Negative one	−1	−1	Additive inverse of 1	300 to 200 BCE	✓	✓	✓	✓
Cube root of 2, Delian constant	${\sqrt[{3}]{2}}$	1.25992 10498 94873 16476 ^{[Mw 6]}^{[OEIS 8]}	Real root of $x^{3}=2$	46 to 120 CE^[11]	✗	✓	✓	✓
Cube root of 3	${\sqrt[{3}]{3}}$	1.44224 95703 07408 38232 ^{[OEIS 9]}	Real root of $x^{3}=3$		✗	✓	✓	✓
Twelfth root of 2^[12]	${\sqrt[{12}]{2}}$	1.05946 30943 59295 26456 ^{[OEIS 10]}	Real positive root of $x^{12}=2$		✗	✓	✓	✓
Supergolden ratio^[13]	$\psi$	1.46557 12318 76768 02665 ^{[OEIS 11]}	${\frac {1+{\sqrt[{3}]{\frac {29+3{\sqrt {93}}}{2}}}+{\sqrt[{3}]{\frac {29-3{\sqrt {93}}}{2}}}}{3}}$ Real root of $x^{3}=x^{2}+1$		✗	✓	✓	✓
Imaginary unit^[14]	$i$	$0 + 1 i$	Principal root of $x^{2}=-1$ ^{[nb 1]}	1501 to 1576	✗	✓	✓	✓
Connective constant for the hexagonal lattice^[15]^[16]	$\mu$	1.84775 90650 22573 51225 ^{[Mw 7]}^{[OEIS 12]}	${\sqrt {2+{\sqrt {2}}}}$ , as a root of the polynomial $x^{4}-4x^{2}+2=0$	1593 ^{[OEIS 12]}	✗	✓	✓	✓
Kepler–Bouwkamp constant^[17]	$K'$	0.11494 20448 53296 20070 ^{[Mw 8]}^{[OEIS 13]}	$\prod _{n=3}^{\infty }\cos \left({\frac {\pi }{n}}\right)=\cos \left({\frac {\pi }{3}}\right)\cos \left({\frac {\pi }{4}}\right)\cos \left({\frac {\pi }{5}}\right)...$	1596 ^{[OEIS 13]}	?	?	?	?
Wallis's constant		2.09455 14815 42326 59148 ^{[Mw 9]}^{[OEIS 14]}	${\sqrt[{3}]{\frac {45-{\sqrt {1929}}}{18}}}+{\sqrt[{3}]{\frac {45+{\sqrt {1929}}}{18}}}$ Real root of $x^{3}-2x-5=0$	1616 to 1703	✗	✓	✓	✓
Euler's number^[18]	$e$	2.71828 18284 59045 23536 ^{[Mw 10]}^{[OEIS 15]}	$\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}=\sum _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}\cdots$	1618^[19]	✗	✗	?	✓
Natural logarithm of 2^[20]	$\ln 2$	0.69314 71805 59945 30941 ^{[Mw 11]}^{[OEIS 16]}	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^[21] & 1668 ^[22]	✗	✗	✓	✓
Sophomore's dream second constant	$I_{2}$	1.29128 59970 62663 54040^{[Mw 12]}^{[OEIS 17]}	$\int _{0}^{1}x^{-x}dx=\sum _{n=1}^{\infty }n^{-n}$	1697	?	?	?	?
Lemniscate constant^[23]	$\varpi$	2.62205 75542 92119 81046 ^{[Mw 13]}^{[OEIS 18]}	$2\int _{0}^{1}{\frac {dt}{\sqrt {1-t^{4}}}}={\frac {1}{4}}{\sqrt {\frac {2}{\pi }}}\,\Gamma {\left({\frac {1}{4}}\right)^{2}}$ Ratio of the perimeter of Bernoulli's lemniscate to its diameter	1718 to 1798	✗	✗	✓	✓
Basel's number	$\zeta (2)$	1.64493 40668 48226 43647 ^{[Mw 14]}^{[OEIS 19]}	$\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+{\frac {1}{5^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}$	1734	✗	✗	✓	✓
Euler's constant	$\gamma$	0.57721 56649 01532 86060 ^{[Mw 15]}^{[OEIS 20]}	$\lim _{n\to \infty }\left(-\log n+\sum _{k=1}^{n}{\frac {1}{k}}\right)=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,dx$ Limiting difference between the harmonic series and the natural logarithm	1735	?	?	?	✓
Erdős–Borwein constant^[24]	$E$	1.60669 51524 15291 76378 ^{[Mw 16]}^{[OEIS 21]}	$\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}={\frac {1}{1}}\!+\!{\frac {1}{3}}\!+\!{\frac {1}{7}}\!+\!{\frac {1}{15}}\!+\!\cdots$	1749^[25]	✗	?	?	?
Square super-root of 2	${\sqrt {2}}_{s}$	1.55961 04694 62369 34997^{[OEIS 22]}	Real root of $x^{x}=2$	1758 & 1783	✗	✗	?	?
Omega constant	$\Omega$	0.56714 32904 09783 87299 ^{[Mw 17]}^{[OEIS 23]}	$W_{0}(1)={\frac {1}{\pi }}\int _{0}^{\pi }\log \left(1+{\frac {\sin t}{t}}e^{t\cot t}\right)dt$ with the Lambert W function $W_{0}$	1758 & 1783	✗	✗	?	?
Apéry's constant^[26]	$\zeta (3)$	1.20205 69031 59594 28539 ^{[Mw 18]}^{[OEIS 24]}	$\zeta (3)=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+\cdots$ with the Riemann zeta function $\zeta (s)$	1780 ^{[OEIS 24]}	✗	?	✓	✓
Laplace limit^[27]		0.66274 34193 49181 58097 ^{[Mw 19]}^{[OEIS 25]}	Real root of ${\frac {xe^{\sqrt {x^{2}+1}}}{{\sqrt {x^{2}+1}}+1}}=1$	~1782	✗	✗	?	?
Soldner constant^[28]^[29]	$\mu$	1.45136 92348 83381 05028 ^{[Mw 20]}^{[OEIS 26]}	$\mathrm {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}=0$ ; Root of the logarithmic integral function	1792 ^{[OEIS 26]}	?	?	?	?
Gauss's constant^[30]	$G$	0.83462 68416 74073 18628 ^{[Mw 21]}^{[OEIS 27]}	${\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}={\frac {1}{4\pi }}{\sqrt {\frac {2}{\pi }}}\Gamma \left({\frac {1}{4}}\right)^{2}={\frac {\varpi }{\pi }}$ where agm is the arithmetic–geometric mean and $\varpi$ is the lemniscate constant	1799^[31]	✗	✗	?	✓
Second Hermite constant^[32]	$\gamma _{2}$	1.15470 05383 79251 52901 ^{[Mw 22]}^{[OEIS 28]}	${\frac {2}{\sqrt {3}}}$	1822 to 1901	✗	✓	✓	✓
Liouville's constant^[33]	$L$	0.11000 10000 00000 00000 0001 ^{[Mw 23]}^{[OEIS 29]}	$\sum _{n=1}^{\infty }{\frac {1}{10^{n!}}}={\frac {1}{10^{1!}}}+{\frac {1}{10^{2!}}}+{\frac {1}{10^{3!}}}+{\frac {1}{10^{4!}}}+\cdots$	Before 1844	✗	✗	?	?
First continued fraction constant	$C_{1}$	0.69777 46579 64007 98201 ^{[Mw 24]}^{[OEIS 30]}	$C_{1}=[0;1,2,3,4,5,...]={\frac {I_{1}(2)}{I_{0}(2)}}$ where I_α(x) are the modified Bessel functions of the first kind^{[nb 2]}	1855^[35]	✗	✗	?	?
Ramanujan's constant^[36]		262 53741 26407 68743 .99999 99999 99250 073 ^{[Mw 25]}^{[OEIS 31]}	$e^{\pi {\sqrt {163}}}$	1859	✗	✗	?	?
Glaisher–Kinkelin constant	$A$	1.28242 71291 00622 63687 ^{[Mw 26]}^{[OEIS 32]}	$e^{{\frac {1}{12}}-\zeta ^{\prime }(-1)}=e^{{\frac {1}{8}}-{\frac {1}{2}}\sum \limits _{n=0}^{\infty }{\frac {1}{n+1}}\sum \limits _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}\left(k+1\right)^{2}\ln(k+1)}$	1860 ^{[OEIS 32]}	?	?	?	?
Catalan's constant^[37]^[38]^[39]	$G$	0.91596 55941 77219 01505 ^{[Mw 27]}^{[OEIS 33]}	$\beta (2)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+{\frac {1}{9^{2}}}+\cdots$ with the Dirichlet beta function $\beta (s)$	1864	?	?	✓	✓
Dottie number^[40]	$D$	0.73908 51332 15160 64165 ^{[Mw 28]}^{[OEIS 34]}	Real root of $\cos x=x$	1865 ^{[Mw 28]}	✗	✗	?	?
Meissel–Mertens constant^[41]	$M$	0.26149 72128 47642 78375 ^{[Mw 29]}^{[OEIS 35]}	$\lim _{n\to \infty }\left(\sum _{p\leq n}{\frac {1}{p}}-\ln \ln n\right)=\gamma +\sum _{p}\left(\ln \left(1-{\frac {1}{p}}\right)+{\frac {1}{p}}\right)$ where γ is the Euler–Mascheroni constant and p is prime	1866 & 1873	?	?	?	?
Universal parabolic constant^[42]	$P$	2.29558 71493 92638 07403 ^{[Mw 30]}^{[OEIS 36]}	$\ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arsinh} (1)+{\sqrt {2}}$	Before 1891^[43]	✗	✗	✓	✓
Cahen's constant^[44]	$C$	0.64341 05462 88338 02618 ^{[Mw 31]}^{[OEIS 37]}	$\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{s_{k}-1}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{6}}-{\frac {1}{42}}+{\frac {1}{1806}}{\,\pm \cdots }$ where s_k is the kth term of Sylvester's sequence (2, 3, 7, 43, 1807, ...)	1891	✗	✗	?	?
Gelfond's constant^[45]	$e^{\pi }$	23.14069 26327 79269 0057 ^{[Mw 32]}^{[OEIS 38]}	$(-1)^{-i}=i^{-2i}=\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}=1+{\frac {\pi ^{1}}{1}}+{\frac {\pi ^{2}}{2}}+{\frac {\pi ^{3}}{6}}+\cdots$	1900^[46]	✗	✗	?	?
Totient summatory constant		1.33978 41535 74347 24659^{[OEIS 39]}	$\prod _{p\in \mathbb {P} }\left(1+{\frac {1}{p^{2}(p-1)}}\right)$ where $\mathbb {P}$ is the set of prime numbers	1900	?	?	?	?
Gelfond–Schneider constant^[47]	$2^{\sqrt {2}}$	2.66514 41426 90225 18865 ^{[Mw 33]}^{[OEIS 40]}	$2^{\sqrt {2}}$	Before 1902 ^{[OEIS 40]}	✗	✗	?	?
Second Favard constant^[48]	$K_{2}$	1.23370 05501 36169 82735 ^{[Mw 34]}^{[OEIS 41]}	${\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots$	1902 to 1965	✗	✗	✓	✓
Golden angle^[49]	$g$	2.39996 32297 28653 32223 ^{[Mw 35]}^{[OEIS 42]}	${\frac {2\pi }{\varphi ^{2}}}=\pi (3-{\sqrt {5}})$ or $180(3-{\sqrt {5}})=137.50776\ldots$ in degrees	1907	✗	✗	✓	✓
Sierpiński's constant^[50]	$K$	2.58498 17595 79253 21706 ^{[Mw 36]}^{[OEIS 43]}	${\begin{aligned}&\pi \left(2\gamma +\ln {\frac {4\pi ^{3}}{\Gamma ({\tfrac {1}{4}})^{4}}}\right)=\pi (2\gamma +4\ln \Gamma ({\tfrac {3}{4}})-\ln \pi )\\&=\pi \left(2\ln 2+3\ln \pi +2\gamma -4\ln \Gamma ({\tfrac {1}{4}})\right)\end{aligned}}$	1907	?	?	?	?
Landau–Ramanujan constant^[51]	$K$	0.76422 36535 89220 66299 ^{[Mw 37]}^{[OEIS 44]}	${\frac {1}{\sqrt {2}}}\prod _{{p\equiv 3{\text{ mod }}4} \atop p\;{\rm {prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{-{\frac {1}{2}}}}\!\!={\frac {\pi }{4}}\prod _{{p\equiv 1{\text{ mod }}4} \atop p\;{\rm {prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{\frac {1}{2}}}$	1908 ^{[OEIS 44]}	?	?	?	?
First Nielsen–Ramanujan constant^[52]	$a_{1}$	0.82246 70334 24113 21823 ^{[Mw 38]}^{[OEIS 45]}	${\frac {{\zeta }(2)}{2}}={\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}\cdots$	1909	✗	✗	✓	✓
Gieseking constant^[53]	$V$	1.01494 16064 09653 62502 ^{[Mw 39]}^{[OEIS 46]}	${\frac {3{\sqrt {3}}}{4}}\left(1-\sum _{n=0}^{\infty }{\frac {1}{(3n+2)^{2}}}+\sum _{n=1}^{\infty }{\frac {1}{(3n+1)^{2}}}\right)$ $={\frac {\sqrt {3}}{3}}\left({\frac {\psi _{1}(1/3)}{2}}-{\frac {\pi ^{2}}{3}}\right)$ with the trigamma function $\psi _{1}$	1912	?	?	✓	✓
Bernstein's constant^[54]	$\beta$	0.28016 94990 23869 13303 ^{[Mw 40]}^{[OEIS 47]}	$\lim _{n\to \infty }2nE_{2n}(f)$ , where E_n(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	?	?	?	?
Tribonacci constant^[55]	$\rho$	1.83928 67552 14161 13255 ^{[Mw 41]}^{[OEIS 48]}	${\textstyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}={\frac {1+4\cosh \left({\frac {1}{3}}\cosh ^{-1}\left(2+{\frac {3}{8}}\right)\right)}{3}}}$ Real root of $x^{3}-x^{2}-x-1=0$	1914 to 1963	✗	✓	✓	✓
Kempner series's value		22.92067 66192 64150 34816^{[OEIS 49]}	$\sum _{n\in N}{\frac {1}{n}}$ where $N$ denotes the set of all positive integers whose decimal expression contains no nines	1914	?	?	?	?
Kempner number	$\kappa$	0.81642 15090 21893 14370^{[OEIS 50]}	$2^{-2^{0}}+2^{-2^{1}}+2^{-2^{2}}+2^{-2^{3}}+\cdots =\sum _{n\geq 0}2^{-2^{n}}$	1916	✗	✗	?	?
Brun's constant^[56]	$B_{2}$	≈ 1.90216 05831 04 ^{[Mw 42]}^{[OEIS 51]}	$\textstyle {\sum \limits _{p}({\frac {1}{p}}+{\frac {1}{p+2}})}=({\frac {1}{3}}\!+\!{\frac {1}{5}})+({\tfrac {1}{5}}\!+\!{\tfrac {1}{7}})+({\tfrac {1}{11}}\!+\!{\tfrac {1}{13}})+\cdots$ where the sum ranges over all primes p such that p + 2 is also a prime	1919 ^{[OEIS 51]}	?	?	?	?
Twin primes constant	$C_{2}$	0.66016 18158 46869 57392 ^{[Mw 43]}^{[OEIS 52]}	$\prod _{\textstyle {p\;{\rm {prime}} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)$	1922	?	?	?	?
Plastic ratio^[57]	$\rho$	1.32471 79572 44746 02596 ^{[Mw 44]}^{[OEIS 53]}	${\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\cdots }}}}}}=\textstyle {\sqrt[{3}]{{\frac {1}{2}}+{\frac {\sqrt {69}}{18}}}}+{\sqrt[{3}]{{\frac {1}{2}}-{\frac {\sqrt {69}}{18}}}}$ Real root of $x^{3}=x+1$	1924 ^{[OEIS 53]}	✗	✓	✓	✓
Bloch's constant^[58]	$B$	$0.4332\leq B\leq 0.4719$ ^{[Mw 45]}^{[OEIS 54]}	The best known bounds are ${\frac {\sqrt {3}}{4}}+2\times 10^{-4}\leq B\leq {\sqrt {\frac {{\sqrt {3}}-1}{2}}}\cdot {\frac {\Gamma ({\frac {1}{3}})\Gamma ({\frac {11}{12}})}{\Gamma ({\frac {1}{4}})}}$	1925 ^{[OEIS 54]}	?	?	?	?
Z score for the 97.5 percentile point^[59]^[60]^[61]^[62]	$z_{.975}$	1.95996 39845 40054 23552 ^{[Mw 46]}^{[OEIS 55]}	${\sqrt {2}}\operatorname {erf} ^{-1}(0.95)$ 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	?	?	?	?
Landau's constant^[58]	$L$	$0.5<L\leq 0.54326$ ^{[Mw 47]}^{[OEIS 56]}	The best known bounds are $0.5<L\leq {\frac {\Gamma ({\frac {1}{3}})\Gamma ({\frac {5}{6}})}{\Gamma ({\frac {1}{6}})}}$ .	1929	?	?	?	?
Landau's third constant^[58]	$A$	$0.5<A\leq 0.7853$		1929	?	?	?	?
Prouhet–Thue–Morse constant^[63]	$\tau$	0.41245 40336 40107 59778 ^{[Mw 48]}^{[OEIS 57]}	$\sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}={\frac {1}{4}}\left[2-\prod _{n=0}^{\infty }\left(1-{\frac {1}{2^{2^{n}}}}\right)\right]$ where ${t_{n}}$ is the n^th term of the Thue–Morse sequence	1929 ^{[OEIS 57]}	✗	✗	?	?
Golomb–Dickman constant^[64]	$\lambda$	0.62432 99885 43550 87099 ^{[Mw 49]}^{[OEIS 58]}	$\int _{0}^{1}e^{\mathrm {li} (t)}dt=\int _{0}^{\infty }{\frac {\rho (t)}{t+2}}dt$ where li(t) is the logarithmic integral, and ρ(t) is the Dickman function	1930 & 1964	?	?	?	?
Constant related to the asymptotic behavior of Lebesgue constants^[65]	$c$	0.98943 12738 31146 95174 ^{[Mw 50]}^{[OEIS 59]}	$\lim _{n\to \infty }\!\!\left(\!{L_{n}{-}{\frac {4}{\pi ^{2}}}\ln(2n{+}1)}\!\!\right)\!{=}{\frac {4}{\pi ^{2}}}\!\left({-}{\frac {\Gamma '({\tfrac {1}{2}})}{\Gamma ({\tfrac {1}{2}})}}{+}{\sum _{k=1}^{\infty }\!{\frac {2\ln k}{4k^{2}{-}1}}}\right)$	1930 ^{[Mw 50]}	?	?	?	?
Feller–Tornier constant^[66]	${\mathcal {C}}_{\mathrm {FT} }$	0.66131 70494 69622 33528 ^{[Mw 51]}^{[OEIS 60]}	${{\frac {1}{2}}\prod _{p{\text{ prime}}}\left(1-{\frac {2}{p^{2}}}\right)+{\frac {1}{2}}}={\frac {3}{\pi ^{2}}}\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p^{2}-1}}\right)+{\frac {1}{2}}$	1932	?	?	?	?
Base 10 Champernowne constant^[67]	$C_{10}$	0.12345 67891 01112 13141 ^{[Mw 52]}^{[OEIS 61]}	Defined by concatenating representations of successive integers: 0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...	1933	✗	✗	?	?
Salem constant^[68]	$\sigma _{10}$	1.17628 08182 59917 50654 ^{[Mw 53]}^{[OEIS 62]}	Largest real root of $x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1=0$	1933 ^{[OEIS 62]}	✗	✓	✓	✓
Khinchin's constant^[69]	$K_{0}$	2.68545 20010 65306 44530 ^{[Mw 54]}^{[OEIS 63]}	$\prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\log _{2}(n)}$	1934	?	?	?	?
Lévy's constant (1)^[70]	$\beta$	1.18656 91104 15625 45282 ^{[Mw 55]}^{[OEIS 64]}	${\frac {\pi ^{2}}{12\,\ln 2}}$	1935	?	?	?	?
Lévy's constant (2)^[71]	$e^{\beta }$	3.27582 29187 21811 15978 ^{[Mw 56]}^{[OEIS 65]}	$e^{\pi ^{2}/(12\ln 2)}$	1936	?	?	?	?
Linnik's constant	$L$	$L$ ≤ 3.99	Exponent constant appearing in the Linnik's theorem about prime numbers in arithmetic progression	1944	?	?	?	?
Copeland–Erdős constant^[72]	${\mathcal {C}}_{CE}$	0.23571 11317 19232 93137 ^{[Mw 57]}^{[OEIS 66]}	Defined by concatenating representations of successive prime numbers: 0.2 3 5 7 11 13 17 19 23 29 31 37 ...	1946 ^{[OEIS 66]}	✗	?	?	?
Mills' constant^[73]	$A$	1.30637 78838 63080 69046^{[nb 3]} ^{[Mw 58]}^{[OEIS 67]}	Smallest positive real number A such that $\lfloor A^{3^{n}}\rfloor$ is prime for all positive integers n^[74]	1947	✗	?	?	?
Gompertz constant^[75]	$\delta$	0.59634 73623 23194 07434 ^{[Mw 59]}^{[OEIS 68]}	$\int _{0}^{\infty }\!\!{\frac {e^{-x}}{1+x}}\,dx=\!\!\int _{0}^{1}\!\!{\frac {dx}{1-\ln x}}={\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {2}{1+{\tfrac {2}{1+{\tfrac {3}{1+3{/\cdots }}}}}}}}}}}}}$	Before 1948 ^{[OEIS 68]}	?	?	?	✓
Third continued fraction constant	$C_{3}$	1.52513 52761 60981 20908^{[Mw 60]}^{[OEIS 69]}	${1+{\tfrac {1}{1+{\tfrac {2}{1+{\tfrac {3}{1+{\tfrac {4}{1+{\tfrac {5}{1+6{/\cdots }}}}}}}}}}}}={\sqrt {\frac {2}{e\pi }}}\left[1-\operatorname {erf} \left({\sqrt {\frac {1}{2}}}\right)\right]^{-1}$ where $\operatorname {erf} (x)$ is the error function	1948	?	?	?	?
de Bruijn–Newman constant	$\Lambda$	$0\leq \Lambda \leq 0.2$	The number $\Lambda$ such that $H(\lambda ,z)=\int _{0}^{\infty }e^{\lambda u^{2}}\Phi (u)\cos(zu)du$ 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 ^{[OEIS 70]}	${\frac {\pi }{\ln 2}}$	Before 1958 ^{[OEIS 71]}	✗	?	?	?
Magic angle^[76]	$\theta _{\mathrm {m} }$	0.95531 66181 245092 78163 ^{[OEIS 72]}	$\arctan {\sqrt {2}}=\arccos {\tfrac {1}{\sqrt {3}}}\approx \textstyle {54.7356}^{\circ }$	Before 1959 ^[77]^[76]	✗	✗	✓	✓
Artin's constant^[78]	$C_{\mathrm {Artin} }$	0.37395 58136 19202 28805 ^{[Mw 61]}^{[OEIS 73]}	$\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p(p-1)}}\right)$	Before 1961 ^{[OEIS 73]}	?	?	?	?
Porter's constant^[79]	$C$	1.46707 80794 33975 47289 ^{[Mw 62]}^{[OEIS 74]}	${\frac {6\ln 2}{\pi ^{2}}}\left(3\ln 2+4\,\gamma -{\frac {24}{\pi ^{2}}}\,\zeta '(2)-2\right)-{\frac {1}{2}}$ where γ is the Euler–Mascheroni constant and $ζ'(2)$ is the derivative of the Riemann zeta function evaluated at s = 2	1961 ^{[OEIS 74]}	?	?	?	?
Lochs constant^[80]	$L$	0.97027 01143 92033 92574 ^{[Mw 63]}^{[OEIS 75]}	${\frac {6\ln 2\ln 10}{\pi ^{2}}}$	1964	?	?	?	?
Infinite tetration of imaginary unit		$0.438283 + 0.360592 i$ ^{[Mw 64]}^{[OEIS 76]}^{[OEIS 77]}	${\frac {2i}{\pi }}W({\frac {-i\pi }{2}})$ where $W$ is the Lambert W function	1966	✗	?	?	?
DeVicci's tesseract constant		1.00743 47568 84279 37609 ^{[OEIS 78]}	Positive root of $4x^{8}{-}28x^{6}{-}7x^{4}{+}16x^{2}{+}16=0$ The edge length of the largest cube that can pass through a 4D hypercube	1966 ^{[OEIS 78]}	✗	✓	✓	✓
Lieb's square ice constant^[81]		1.53960 07178 39002 03869 ^{[Mw 65]}^{[OEIS 79]}	$\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}$	1967	✗	✓	✓	✓
Growth rate constant of fixed polyomino (Klarner's constant)^[82]	$\lambda _{2}$	4.00253 < $\lambda _{2}$ < 4.5252	$\lim _{n\rightarrow \infty }(M_{n})^{\frac {1}{n}}$ where $M_{n}$ is the number of fixed polyominoes of size n	1967	?	?	?	?
Growth rate constant of fixed polycube^[82]	$\lambda _{3}$	6.3795 < $\lambda _{3}$ < 9.3835	$\lim _{n\rightarrow \infty }(M_{n})^{\frac {1}{n}}$ where $M_{n}$ is the number of fixed polycubes of size n	1967	?	?	?	?
Niven's constant^[83]	$C$	1.70521 11401 05367 76428 ^{[Mw 66]}^{[OEIS 80]}	$1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)}}\right)$	1969	?	?	?	?
Stephens' constant^[84]	$C_{S}$	0.57595 99688 92945 43964 ^{[Mw 67]}^{[OEIS 81]}	$\prod _{p{\text{ prime}}}\left(1-{\frac {p}{p^{3}-1}}\right)$	1969 ^{[OEIS 81]}	?	?	?	?
Regular paperfolding sequence^[85]^[86]	$P$	0.85073 61882 01867 26036 ^{[Mw 68]}^{[OEIS 82]}	$\sum _{n=0}^{\infty }{\frac {8^{2^{n}}}{2^{2^{n+2}}-1}}=\sum _{n=0}^{\infty }{\cfrac {\tfrac {1}{2^{2^{n}}}}{1-{\tfrac {1}{2^{2^{n+2}}}}}}$	1970 ^{[OEIS 82]}	✗	✗	?	?
Reciprocal Fibonacci constant^[87]	$\psi$	3.35988 56662 43177 55317 ^{[Mw 69]}^{[OEIS 83]}	$\sum _{n=1}^{\infty }{\frac {1}{F_{n}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+\cdots$ where F_n is the n^th Fibonacci number	1974 ^{[OEIS 83]}	✗	?	?	?
Chvátal–Sankoff constant for the binary alphabet	$\gamma _{2}$	$0.788071\leq \gamma _{2}\leq 0.826280$	$\lim _{n\to \infty }{\frac {\operatorname {E} [\lambda _{n,2}]}{n}}$ where $E[λ n,2]$ is the expected longest common subsequence of two random length-n binary strings	1975	?	?	?	?
Feigenbaum constant δ^[88]	$\delta$	4.66920 16091 02990 67185 ^{[Mw 70]}^{[OEIS 84]}	$\lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{a_{n+2}-a_{n+1}}}$ where the sequence a_n is given by n-th period-doubling bifurcation of logistic map $x_{k+1}=ax_{k}(1-x_{k})$ or any other one-dimensional map with a single quadratic maximum	1975	?	?	?	?
Chaitin's constants^[89]	$\Omega$	In general they are uncomputables; one such number is 0.00787 49969 97812 3844. ^{[Mw 71]}^{[OEIS 85]}	$\sum _{p\in P}2^{-\|p\|}$ where $p$ is a halted program, $\| p \|$ is its size in bits, and $P$ is the domain of all programs that stop	1975	✗	✗	✗	✗
Freiman's constant	$F$	4.52782 95661 60879 14088^{[OEIS 86]}	End constant of the last gap in the Lagrange spectrum, a proper subset of the Markov spectrum	1975	✗	✓	✓	✓
Robbins constant^[90]	$\Delta (3)$	0.66170 71822 67176 23515 ^{[Mw 72]}^{[OEIS 87]}	${\frac {4\!+\!17{\sqrt {2}}\!-6{\sqrt {3}}\!-7\pi }{105}}\!+\!{\frac {\ln(1\!+\!{\sqrt {2}})}{5}}\!+\!{\frac {2\ln(2\!+\!{\sqrt {3}})}{5}}$	1977 ^{[OEIS 87]}	✗	✗	✓	✓
Weierstrass constant^[91]		0.47494 93799 87920 65033 ^{[Mw 73]}^{[OEIS 88]}	${\frac {2^{5/4}{\sqrt {\pi }}\,e^{\pi /8}}{\Gamma ({\frac {1}{4}})^{2}}}$ ${\frac {\operatorname {\sigma } {(1;1,i)}}{2}}$ , where σ(z;ω₁,ω₂) is the Weierstrass sigma function with half-periods ω₁ and ω₂	Before 1978^[92]	✗	✗	?	?
Fransén–Robinson constant^[93]	$F$	2.80777 02420 28519 36522 ^{[Mw 74]}^{[OEIS 89]}	$\int _{0}^{\infty }{\frac {dx}{\Gamma (x)}}=e+\int _{0}^{\infty }{\frac {e^{-x}}{\pi ^{2}+\ln ^{2}x}}\,dx$	1978	?	?	?	?
Feigenbaum constant α^[94]	$\alpha$	2.50290 78750 95892 82228 ^{[Mw 70]}^{[OEIS 90]}	Ratio between the width of a tine and the width of one of its two subtines in a bifurcation diagram	1979	?	?	?	?
Beam detection constant^[95]	$L$	3.1415926535 ≤ $L$ ≤ 4.7998	Length of the shortest opaque set for the unit circle	Before 1980	?	?	?	?
Irrational number from Moser–de Bruijn sequence^[96]	$\alpha$	3.30033 00000 00000 33003^{[OEIS 91]}	$3*\sum _{n\in E}10^{-n}$ where $E$ denotes the Moser–de Bruijn sequence	1982	✗	✗	?	?
Second du Bois-Reymond constant^[97]	$C_{2}$	0.19452 80494 65325 11361 ^{[Mw 75]}^{[OEIS 92]}	${\frac {e^{2}-7}{2}}=\int _{0}^{\infty }\left\|{{\frac {d}{dt}}\left({\frac {\sin t}{t}}\right)^{2}}\right\|\,dt-1$	1983 ^{[OEIS 92]}	✗	✗	?	✓
Erdős–Tenenbaum–Ford constant	$\delta$	0.08607 13320 55934 20688 ^{[OEIS 93]}	$1-{\frac {1+\ln \ln 2}{\ln 2}}$	1984	?	?	?	?
Conway's constant^[98]	$\lambda$	1.30357 72690 34296 39125 ^{[Mw 76]}^{[OEIS 94]}	Real root of the polynomial ${\begin{smallmatrix}x^{71}-x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}-x^{60}\\-x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{50}\\+x^{49}+9x^{48}-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{40}\\-12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{30}\\-10x^{29}-3x^{28}+2x^{27}+9x^{26}-3x^{25}+14x^{24}-8x^{23}-7x^{21}+9x^{20}\\+3x^{19}\!-4x^{18}\!-10x^{17}\!-7x^{16}\!+12x^{15}\!+7x^{14}\!+2x^{13}\!-12x^{12}\!-4x^{11}\!-2x^{10}\\+5x^{9}+x^{7}-7x^{6}+7x^{5}-4x^{4}+12x^{3}-6x^{2}+3x-6\ =\ 0\quad \quad \quad \end{smallmatrix}}$ Limiting growth rate of the Look-and-say sequence starting at 1	1987	✗	✓	✓	✓
Hard hexagon entropy constant	$\kappa (1)$	1.39548 59724 79302 73522^{[Mw 77]}^{[OEIS 95]}	Positive real root of the 24th degree polynomial^{[nb 4]}	1988	✗	✓	✓	✓
Hafner–Sarnak–McCurley constant^[99]	$\sigma$	0.35323 63718 54995 98454 ^{[Mw 78]}^{[OEIS 96]}	$\prod _{p{\text{ prime}}}{\left(1-\left(1-\prod _{n\geq 1}\left(1-{\frac {1}{p^{n}}}\right)\right)^{2}\right)}\!$	1991 ^{[OEIS 96]}	?	?	?	?
Backhouse's constant^[100]	$B$	1.45607 49485 82689 67139 ^{[Mw 79]}^{[OEIS 97]}	$\lim _{k\to \infty }\left\|{\frac {q_{k+1}}{q_{k}}}\right\vert \quad \scriptstyle {\text{where:}}\displaystyle \;\;Q(x)={\frac {1}{P(x)}}=\!\sum _{k=1}^{\infty }q_{k}x^{k}$ $P(x)=1+\sum _{k=1}^{\infty }{p_{k}x^{k}}=1+2x+3x^{2}+5x^{3}+\cdots$ where p_k is the k^th prime number	1995	?	?	?	?
Viswanath constant^[101]	$V$	1.13198 82487 943 ^{[Mw 80]}^{[OEIS 98]}	$\lim _{n\to \infty }\|f_{n}\|^{\frac {1}{n}}$ where f_n = f_n−1 ± f_n−2, where the signs + or − are chosen at random with equal probability 1/2	1997	?	?	?	?
Komornik–Loreti constant^[102]	$q$	1.78723 16501 82965 93301 ^{[Mw 81]}^{[OEIS 99]}	Real number $q$ such that $1=\sum _{k=1}^{\infty }{\frac {t_{k}}{q^{k}}}$ , or $\prod _{n=0}^{\infty }\left(1-{\frac {1}{q^{2^{n}}}}\right)+{\frac {q-2}{q-1}}=0$ where t_k is the k^th term of the Thue–Morse sequence	1998	✗	✗	?	?
Natural density constant of abundant numbers	$d_{A}$	0.2476171 < $d_{A}$ < 0.2476475^[103]^[104]	$\lim _{n\rightarrow \infty }{\frac {\beta (n)}{n}}$ where $\beta (n)$ is the number of abundant numbers less than or equal to n	Before 1998	?	?	?	?
Embree–Trefethen constant	$\beta ^{\star }$	0.70258	The smallest value β such that solutions to the random sequence ${\textstyle f_{n}=\pm f_{n-1}\pm \beta f_{n-2}}$ do not decay	1999	?	?	?	?
Heath-Brown–Moroz constant^[105]	$C$	0.00131 76411 54853 17810 ^{[Mw 82]}^{[OEIS 100]}	$\prod _{p{\text{ prime}}}\left(1-{\frac {1}{p}}\right)^{7}\left(1+{\frac {7p+1}{p^{2}}}\right)$	1999 ^{[OEIS 100]}	?	?	?	?
MRB constant^[106]^[107]^[108]	$S$	0.18785 96424 62067 12024 ^{[Mw 83]}^{[Ow 1]}^{[OEIS 101]}	$\sum _{n=1}^{\infty }(-1)^{n}(n^{1/n}-1)=-{\sqrt[{1}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+\cdots$	1999	?	?	?	?
Prime constant^[109]	$\rho$	0.41468 25098 51111 66024 ^{[OEIS 102]}	$\sum _{p{\text{ prime}}}{\frac {1}{2^{p}}}={\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{32}}+\cdots$	1999 ^{[OEIS 102]}	✗	?	?	?
Somos' quadratic recurrence constant^[110]	$\sigma$	1.66168 79496 33594 12129 ^{[Mw 84]}^{[OEIS 103]}	$\prod _{n=1}^{\infty }n^{{1/2}^{n}}={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots$	1999 ^{[Mw 84]}	?	?	?	?
Foias constant^[111]	$\alpha$	1.18745 23511 26501 05459 ^{[Mw 85]}^{[OEIS 104]}	The unique real number α such that if x₁ = α then the sequence $x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}{\text{ for }}n=1,2,3,\ldots$ diverges to infinity	2000	?	?	?	?
Logarithmic capacity of the unit disk^[112]^[113]		0.59017 02995 08048 11302^{[Mw 86]}^{[OEIS 105]}	${\frac {\Gamma ({\tfrac {1}{4}})^{2}}{4\pi ^{3/2}}}={\frac {\varpi }{\pi {\sqrt {2}}}}$ where $\varpi$ is the lemniscate constant	Before 2003 ^{[OEIS 105]}	✗	✗	?	✓
Taniguchi constant^[84]	$C_{T}$	0.67823 44919 17391 97803^{[Mw 87]}^{[OEIS 106]}	$\prod _{p{\text{ prime}}}\left(1-{\frac {3}{p^{3}}}+{\frac {2}{p^{4}}}+{\frac {1}{p^{5}}}-{\frac {1}{p^{6}}}\right)$	Before 2005^[84]	?	?	?	?

Name	Symbol	Set	Decimal expansion	Continued fraction	Notes
Zero	0	$\mathbb {Z}$	0.00000 00000	[0; ]
Golomb–Dickman constant	$\lambda$		0.62432 99885	[0; 1, 1, 1, 1, 1, 22, 1, 2, 3, 1, 1, 11, 1, 1, 2, 22, 2, 6, 1, 1, …]^{[OEIS 107]}	Eric Weisstein noted that the continued fraction has an unusually large number of 1s.^{[Mw 88]}
Cahen's constant	$C_{2}$	$\mathbb {R} \setminus \mathbb {A}$	0.64341 05463	[0; 1, 1, 1, 2², 3², 13², 129², 25298², 420984147², 269425140741515486², …]^{[OEIS 108]}	All terms are squares and truncated at 10 terms due to large size. Davison and Shallit used the continued fraction expansion to prove that the constant is transcendental.
Euler–Mascheroni constant	$\gamma$		0.57721 56649^[114]	[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …] ^[114]^{[OEIS 109]}	Using the continued fraction expansion, it was shown that if $γ$ is rational, its denominator must exceed 10²⁴⁴⁶⁶³.
First continued fraction constant	$C_{1}$	$\mathbb {R} \setminus \mathbb {A}$	0.69777 46579	[0; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …]	Equal to the ratio $I_{1}(2)/I_{0}(2)$ of modified Bessel functions of the first kind evaluated at 2.
Catalan's constant	$G$		0.91596 55942^[115]	[0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …] ^[115]^{[OEIS 110]}	Computed up to 4851389025 terms by E. Weisstein.^{[Mw 89]}
One half	⁠1/2⁠	$\mathbb {Q}$	0.50000 00000	[0; 2]
Prouhet–Thue–Morse constant	$\tau$	$\mathbb {R} \setminus \mathbb {A}$	0.41245 40336	[0; 2, 2, 2, 1, 4, 3, 5, 2, 1, 4, 2, 1, 5, 44, 1, 4, 1, 2, 4, 1, …]^{[OEIS 111]}	Infinitely many partial quotients are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.^[116]
Copeland–Erdős constant	${\mathcal {C}}_{CE}$	$\mathbb {R} \setminus \mathbb {Q}$	0.23571 11317	[0; 4, 4, 8, 16, 18, 5, 1, 1, 1, 1, 7, 1, 1, 6, 2, 9, 58, 1, 3, 4, …]^{[OEIS 112]}	Computed up to 1011597392 terms by E. Weisstein. He also noted that while the Champernowne constant continued fraction contains sporadic large terms, the continued fraction of the Copeland–Erdős Constant do not exhibit this property.^{[Mw 90]}
Base 10 Champernowne constant	$C_{10}$	$\mathbb {R} \setminus \mathbb {A}$	0.12345 67891	[0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 4.57540×10¹⁶⁵, 6, 1, …] ^{[OEIS 113]}	Champernowne constants in any base exhibit sporadic large numbers; the 40th term in $C_{10}$ has 2504 digits.
One	1	$\mathbb {N}$	1.00000 00000	[1; ]
Phi, Golden ratio	$\varphi$	$\mathbb {A}$	1.61803 39887^[117]	[1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …] ^[118]	The convergents are ratios of successive Fibonacci numbers.
Brun's constant	$B_{2}$		1.90216 05831	[1; 1, 9, 4, 1, 1, 8, 3, 4, 7, 1, 3, 3, 1, 2, 1, 1, 12, 4, 2, 1, …]	The n^th roots of the denominators of the n^th convergents are close to Khinchin's constant, suggesting that $B_{2}$ is irrational. If true, this would prove the twin prime conjecture.^[119]
Square root of 2	${\sqrt {2}}$	$\mathbb {A}$	1.41421 35624	[1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, …]	The convergents are ratios of successive Pell numbers.
Two	2	$\mathbb {N}$	2.00000 00000	[2; ]
Euler's number	$e$	$\mathbb {R} \setminus \mathbb {A}$	2.71828 18285^[120]	[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …] ^[121]^{[OEIS 114]}	The continued fraction expansion has the pattern [2; 1, 2, 1, 1, 4, 1, ..., 1, 2n, 1, ...].
Khinchin's constant	$K_{0}$		2.68545 20011^[122]	[2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …] ^[123]^{[OEIS 115]}	For almost all real numbers x, the coefficients of the continued fraction of x have a finite geometric mean known as Khinchin's constant.
Three	3	$\mathbb {N}$	3.00000 00000	[3; ]
Pi	$\pi$	$\mathbb {R} \setminus \mathbb {A}$	3.14159 26536	[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, …] ^{[OEIS 116]}	The first few convergents (3, 22/7, 333/106, 355/113, ...) are among the best-known and most widely used historical approximations of $π$ .

Name	Symbol	Formula	Year	Set
Harmonic number	$H_{n}$	$\sum _{k=1}^{n}{\frac {1}{k}}$	Antiquity	$\mathbb {Q}$
Gregory coefficients	$G_{n}$	${\frac {1}{n!}}\int _{0}^{1}x(x-1)(x-2)\cdots (x-n+1)\,dx=\int _{0}^{1}{\binom {x}{n}}\,dx$	1670	$\mathbb {Q}$
Bernoulli number	$B_{n}^{\pm }$	${\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}\pm 1\right)=\sum _{m=0}^{\infty }{\frac {B_{m}^{\pm {}}t^{m}}{m!}}$	1689	$\mathbb {Q}$
Root of unity	$\zeta _{n}$	$\exp({\frac {2k\pi i}{n}})$	Before 1722	$\mathbb {A}$
Hermite constants^{[Mw 91]}	$\gamma _{n}$	For a lattice L in Euclidean space Rⁿ with unit covolume, i.e. vol(Rⁿ/L) = 1, let λ₁(L) denote the least length of a nonzero element of L. Then √γ_nn is the maximum of λ₁(L) over all such lattices L.	1822 to 1901	$\mathbb {R}$
Hafner–Sarnak–McCurley constant^[124]	$D(n)$	$D(n)=\prod _{k=1}^{\infty }\left\{1-\left[1-\prod _{j=1}^{n}(1-p_{k}^{-j})\right]^{2}\right\}$	1883^{[Mw 92]}	$\mathbb {R}$
Stieltjes constants	$\gamma _{n}$	${\frac {(-1)^{n}n!}{2\pi }}\int _{0}^{2\pi }e^{-nix}\zeta \left(e^{ix}+1\right)dx$	before 1894	$\mathbb {R}$
Favard constants^[48]^{[Mw 93]}	$K_{r}$	${\frac {4}{\pi }}\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}}{2n+1}}\right)^{\!r+1}={\frac {4}{\pi }}\left({\frac {(-1)^{0(r+1)}}{1^{r}}}+{\frac {(-1)^{1(r+1)}}{3^{r}}}+{\frac {(-1)^{2(r+1)}}{5^{r}}}+{\frac {(-1)^{3(r+1)}}{7^{r}}}+\cdots \right)$	1902 to 1965	$\mathbb {R}$
Generalized Brun's Constant^[56]	$B_{n}$	${\sum \limits _{p}\left({\frac {1}{p}}+{\frac {1}{p+n}}\right)}$ where the sum ranges over all primes p such that p + n is also a prime	1919^{[OEIS 51]}	$\mathbb {R}$
Champernowne constants^[67]	$C_{b}$	Defined by concatenating representations of successive integers in base b $C_{b}=\sum _{n=1}^{\infty }{\frac {n}{b^{\left(\sum _{k=1}^{n}\lceil \log _{b}(k+1)\rceil \right)}}}$	1933	$\mathbb {R} \setminus \mathbb {A}$
Lagrange number	$L(n)$	${\sqrt {9-{\frac {4}{{m_{n}}^{2}}}}}$ where $m_{n}$ is the nth smallest number such that $m^{2}+x^{2}+y^{2}=3mxy\,$ has positive (x,y)	before 1957	$\mathbb {A}$
Feller's coin-tossing constants	$\alpha _{k},\beta _{k}$	$\alpha _{k}$ is the smallest positive real root of $x^{k+1}=2^{k+1}(x-1),\beta _{k}={\frac {2-\alpha _{k}}{k+1-k\alpha _{k}}}$	1968	$\mathbb {A}$
Stoneham number	$\alpha _{b,c}$	$\sum _{n=c^{k}>1}{\frac {1}{b^{n}n}}=\sum _{k=1}^{\infty }{\frac {1}{b^{c^{k}}c^{k}}}$ where b,c are coprime integers	1973	$\mathbb {R} \setminus \mathbb {A}$
Beraha constants	$B(n)$	$2+2\cos \left({\frac {2\pi }{n}}\right)$	1974	$\mathbb {A}$
Chvátal–Sankoff constants	$\gamma _{k}$	$\lim _{n\to \infty }{\frac {E[\lambda _{n,k}]}{n}}$	1975	$\mathbb {R}$
Hyperharmonic number	$H_{n}^{(r)}$	$\sum _{k=1}^{n}H_{k}^{(r-1)}$ and $H_{n}^{(0)}={\frac {1}{n}}$	1995	$\mathbb {Q}$
Gregory number	$G_{x}$	$\sum _{n=0}^{\infty }(-1)^{n}{\frac {1}{(2n+1)x^{2n+1}}}=\operatorname {arccot}(x)$ for rational x greater than or equal to one	before 1996	$\mathbb {R} \setminus \mathbb {A}$
Metallic mean		${\frac {n+{\sqrt {n^{2}+4}}}{2}}$	before 1998	$\mathbb {A}$

List of mathematical constants

List

Mathematical constants sorted by their representations as continued fractions

Sequences of constants

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