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List of mathematical constants

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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
Q {\displaystyle \mathbb {Q} } A {\displaystyle \mathbb {A} } P {\displaystyle {\mathcal {P}}}
One 1 1 Multiplicative identity of C {\displaystyle \mathbb {C} } . Prehistory ✓ ✓ ✓
Two 2 2 Prehistory ✓ ✓ ✓
One half 1/2 0.5 Prehistory ✓ ✓ ✓
Pi π {\displaystyle \pi } 3.14159 26535 89793 23846 Ratio of a circle's circumference to its diameter. 1900 to 1600 BCE ✗ ✗ ✓
Tau τ {\displaystyle \tau } 6.28318 53071 79586 47692 Ratio of a circle's circumference to its radius. Equal to 2 π {\displaystyle 2\pi } 1900 to 1600 BCE ✗ ✗ ✓
Square root of 2,

Pythagoras constant

2 {\displaystyle {\sqrt {2}}} 1.41421 35623 73095 04880 Positive root of x 2 = 2 {\displaystyle x^{2}=2} 1800 to 1600 BCE ✗ ✓ ✓
Square root of 3,

Theodorus' constant

3 {\displaystyle {\sqrt {3}}} 1.73205 08075 68877 29352 Positive root of x 2 = 3 {\displaystyle x^{2}=3} 465 to 398 BCE ✗ ✓ ✓
Square root of 5 5 {\displaystyle {\sqrt {5}}} 2.23606 79774 99789 69640 Positive root of x 2 = 5 {\displaystyle x^{2}=5} ✗ ✓ ✓
Phi, Golden ratio φ {\displaystyle \varphi } or ϕ {\displaystyle \phi } 1.61803 39887 49894 84820 1 + 5 2 {\displaystyle {\frac {1+{\sqrt {5}}}{2}}} ~300 BCE ✗ ✓ ✓
Silver ratio δ S {\displaystyle \delta _{S}} 2.41421 35623 73095 04880 2 + 1 {\displaystyle {\sqrt {2}}+1} ~300 BCE ✗ ✓ ✓
Zero 0 0 Additive identity of C {\displaystyle \mathbb {C} } . 300 to 100 BCE ✓ ✓ ✓
Negative one −1 −1 300 to 200 BCE ✓ ✓ ✓
Cube root of 2 2 3 {\displaystyle {\sqrt{2}}} 1.25992 10498 94873 16476 Real root of x 3 = 2 {\displaystyle x^{3}=2} 46 to 120 CE ✗ ✓ ✓
Cube root of 3 3 3 {\displaystyle {\sqrt{3}}} 1.44224 95703 07408 38232 Real root of x 3 = 3 {\displaystyle x^{3}=3} ✗ ✓ ✓
Twelfth root of 2 2 12 {\displaystyle {\sqrt{2}}} 1.05946 30943 59295 26456 Real root of x 12 = 2 {\displaystyle x^{12}=2} ✗ ✓ ✓
Supergolden ratio ψ {\displaystyle \psi } 1.46557 12318 76768 02665 1 + 29 + 3 93 2 3 + 29 − 3 93 2 3 3 {\displaystyle {\frac {1+{\sqrt{\frac {29+3{\sqrt {93}}}{2}}}+{\sqrt{\frac {29-3{\sqrt {93}}}{2}}}}{3}}}

Real root of x 3 = x 2 + 1 {\displaystyle x^{3}=x^{2}+1}

✗ ✓ ✓
Imaginary unit i {\displaystyle i} 0 + 1i Principal root of x 2 = − 1 {\displaystyle x^{2}=-1} 1501 to 1576 ✗ ✓ ✓
Connective constant for the hexagonal lattice μ {\displaystyle \mu } 1.84775 90650 22573 51225 2 + 2 {\displaystyle {\sqrt {2+{\sqrt {2}}}}} , as a root of the polynomial x 4 − 4 x 2 + 2 = 0 {\displaystyle x^{4}-4x^{2}+2=0} 1593 ✗ ✓ ✓
Kepler–Bouwkamp constant K ′ {\displaystyle K'} 0.11494 20448 53296 20070 ∏ n = 3 ∞ cos ⁡ ( π n ) = cos ⁡ ( π 3 ) cos ⁡ ( π 4 ) cos ⁡ ( π 5 ) . . . {\displaystyle \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 ? ? ?
Wallis's constant 2.09455 14815 42326 59148 45 − 1929 18 3 + 45 + 1929 18 3 {\displaystyle {\sqrt{\frac {45-{\sqrt {1929}}}{18}}}+{\sqrt{\frac {45+{\sqrt {1929}}}{18}}}}

Real root of x 3 − 2 x − 5 = 0 {\displaystyle x^{3}-2x-5=0}

1616 to 1703 ✗ ✓ ✓
Euler's number e {\displaystyle e} 2.71828 18284 59045 23536 lim n → ∞ ( 1 + 1 n ) n = ∑ n = 0 ∞ 1 n ! = 1 + 1 1 ! + 1 2 ! + 1 3 ! ⋯ {\displaystyle \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 ✗ ✗ ?
Natural logarithm of 2 ln ⁡ 2 {\displaystyle \ln 2} 0.69314 71805 59945 30941 Real root of e x = 2 {\displaystyle e^{x}=2}

∑ n = 1 ∞ ( − 1 ) n + 1 n = 1 1 − 1 2 + 1 3 − 1 4 + ⋯ {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+\cdots }

1619 & 1668 ✗ ✗ ✓
Lemniscate constant ϖ {\displaystyle \varpi } 2.62205 75542 92119 81046 2 ∫ 0 1 d t 1 − t 4 = 1 4 2 π Γ ( 1 4 ) 2 {\displaystyle 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 ✗ ✗ ✓
Euler's constant γ {\displaystyle \gamma } 0.57721 56649 01532 86060 lim n → ∞ ( − log ⁡ n + ∑ k = 1 n 1 k ) = ∫ 1 ∞ ( − 1 x + 1 ⌊ x ⌋ ) d x {\displaystyle \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 E {\displaystyle E} 1.60669 51524 15291 76378 ∑ n = 1 ∞ 1 2 n − 1 = 1 1 + 1 3 + 1 7 + 1 15 + ⋯ {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}={\frac {1}{1}}\!+\!{\frac {1}{3}}\!+\!{\frac {1}{7}}\!+\!{\frac {1}{15}}\!+\!\cdots } 1749 ✗ ? ?
Omega constant Ω {\displaystyle \Omega } 0.56714 32904 09783 87299 W ( 1 ) = 1 π ∫ 0 π log ⁡ ( 1 + sin ⁡ t t e t cot ⁡ t ) d t {\displaystyle 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 ✗ ✗ ?
Apéry's constant ζ ( 3 ) {\displaystyle \zeta (3)} 1.20205 69031 59594 28539 ζ ( 3 ) = ∑ n = 1 ∞ 1 n 3 = 1 1 3 + 1 2 3 + 1 3 3 + 1 4 3 + 1 5 3 + ⋯ {\displaystyle \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 ζ ( s ) {\displaystyle \zeta (s)} .

1780 ✗ ? ✓
Laplace limit 0.66274 34193 49181 58097 Real root of x e x 2 + 1 x 2 + 1 + 1 = 1 {\displaystyle {\frac {xe^{\sqrt {x^{2}+1}}}{{\sqrt {x^{2}+1}}+1}}=1} ~1782 ✗ ✗ ?
Soldner constant μ {\displaystyle \mu } 1.45136 92348 83381 05028 l i ( x ) = ∫ 0 x d t ln ⁡ t = 0 {\displaystyle \mathrm {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}=0} ; root of the logarithmic integral function. 1792 ? ? ?
Gauss's constant G {\displaystyle G} 0.83462 68416 74073 18628 1 a g m ( 1 , 2 ) = 1 4 π 2 π Γ ( 1 4 ) 2 = ϖ π {\displaystyle {\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 ϖ {\displaystyle \varpi } is the lemniscate constant.

1799 ✗ ✗ ?
Second Hermite constant γ 2 {\displaystyle \gamma _{2}} 1.15470 05383 79251 52901 2 3 {\displaystyle {\frac {2}{\sqrt {3}}}} 1822 to 1901 ✗ ✓ ✓
Liouville's constant L {\displaystyle L} 0.11000 10000 00000 00000 0001 ∑ n = 1 ∞ 1 10 n ! = 1 10 1 ! + 1 10 2 ! + 1 10 3 ! + 1 10 4 ! + ⋯ {\displaystyle \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 {\displaystyle C_{1}} 0.69777 46579 64007 98201 C 2 = [ 0 ; 1 , 2 , 3 , 4 , 5 , . . . ] = I 1 ( 2 ) I 0 ( 2 ) {\displaystyle C_{2}=={\frac {I_{1}(2)}{I_{0}(2)}}} , (see Bessel functions). C 2 ∉ A . {\displaystyle C_{2}\notin \mathbb {A} .} 1855 ✗ ✗ ?
Ramanujan's constant 262 53741 26407 68743
.99999 99999 99250 073 		e π 163 {\displaystyle e^{\pi {\sqrt {163}}}} 1859 ✗ ✗ ?
Glaisher–Kinkelin constant A {\displaystyle A} 1.28242 71291 00622 63687 e 1 12 − ζ ′ ( − 1 ) = e 1 8 − 1 2 ∑ n = 0 ∞ 1 n + 1 ∑ k = 0 n ( − 1 ) k ( n k ) ( k + 1 ) 2 ln ⁡ ( k + 1 ) {\displaystyle 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 ? ? ?
Catalan's constant G {\displaystyle G} 0.91596 55941 77219 01505 β ( 2 ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) 2 = 1 1 2 − 1 3 2 + 1 5 2 − 1 7 2 + 1 9 2 + ⋯ {\displaystyle \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 β ( s ) {\displaystyle \beta (s)} .

1864 ? ? ✓
Dottie number 0.73908 51332 15160 64165 Real root of cos ⁡ x = x {\displaystyle \cos x=x} 1865 ✗ ✗ ?
Meissel–Mertens constant M {\displaystyle M} 0.26149 72128 47642 78375 lim n → ∞ ( ∑ p ≤ n 1 p − ln ⁡ ln ⁡ n ) = γ + ∑ p ( ln ⁡ ( 1 − 1 p ) + 1 p ) {\displaystyle \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 P {\displaystyle P} 2.29558 71493 92638 07403 ln ⁡ ( 1 + 2 ) + 2 = arsinh ⁡ ( 1 ) + 2 {\displaystyle \ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arsinh} (1)+{\sqrt {2}}} Before 1891 ✗ ✗ ✓
Cahen's constant C {\displaystyle C} 0.64341 05462 88338 02618 ∑ k = 1 ∞ ( − 1 ) k s k − 1 = 1 1 − 1 2 + 1 6 − 1 42 + 1 1806 ± ⋯ {\displaystyle \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 ✗ ✗ ?
Gelfond's constant e π {\displaystyle e^{\pi }} 23.14069 26327 79269 0057 ( − 1 ) − i = i − 2 i = ∑ n = 0 ∞ π n n ! = 1 + π 1 1 + π 2 2 + π 3 6 + ⋯ {\displaystyle (-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 ✗ ✗ ?
Gelfond–Schneider constant 2 2 {\displaystyle 2^{\sqrt {2}}} 2.66514 41426 90225 18865 2 2 {\displaystyle 2^{\sqrt {2}}} Before 1902 ✗ ✗ ?
Second Favard constant K 2 {\displaystyle K_{2}} 1.23370 05501 36169 82735 π 2 8 = ∑ n = 0 ∞ 1 ( 2 n − 1 ) 2 = 1 1 2 + 1 3 2 + 1 5 2 + 1 7 2 + ⋯ {\displaystyle {\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 g {\displaystyle g} 2.39996 32297 28653 32223 2 π φ 2 = π ( 3 − 5 ) {\displaystyle {\frac {2\pi }{\varphi ^{2}}}=\pi (3-{\sqrt {5}})} or

180 ( 3 − 5 ) = 137.50776 … {\displaystyle 180(3-{\sqrt {5}})=137.50776\ldots } in degrees

1907 ✗ ✗ ✓
Sierpiński's constant K {\displaystyle K} 2.58498 17595 79253 21706 π ( 2 γ + ln ⁡ 4 π 3 Γ ( 1 4 ) 4 ) = π ( 2 γ + 4 ln ⁡ Γ ( 3 4 ) − ln ⁡ π ) = π ( 2 ln ⁡ 2 + 3 ln ⁡ π + 2 γ − 4 ln ⁡ Γ ( 1 4 ) ) {\displaystyle {\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 K {\displaystyle K} 0.76422 36535 89220 66299 1 2 ∏ p ≡ 3  mod  4 p p r i m e ( 1 − 1 p 2 ) − 1 2 = π 4 ∏ p ≡ 1  mod  4 p p r i m e ( 1 − 1 p 2 ) 1 2 {\displaystyle {\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 ? ? ?
First Nielsen–Ramanujan constant a 1 {\displaystyle a_{1}} 0.82246 70334 24113 21823 ζ ( 2 ) 2 = π 2 12 = ∑ n = 1 ∞ ( − 1 ) n + 1 n 2 = 1 1 2 − 1 2 2 + 1 3 2 − 1 4 2 + ⋯ {\displaystyle {\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 G {\displaystyle G} 1.01494 16064 09653 62502 3 3 4 ( 1 − ∑ n = 0 ∞ 1 ( 3 n + 2 ) 2 + ∑ n = 1 ∞ 1 ( 3 n + 1 ) 2 ) {\displaystyle {\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)}
= 3 3 ( ψ 1 ( 1 / 3 ) 2 − π 2 3 ) {\displaystyle ={\frac {\sqrt {3}}{3}}\left({\frac {\psi _{1}(1/3)}{2}}-{\frac {\pi ^{2}}{3}}\right)} with the trigamma function ψ 1 {\displaystyle \psi _{1}} . 		1912 ? ? ✓
Bernstein's constant β {\displaystyle \beta } 0.28016 94990 23869 13303 lim n → ∞ 2 n E 2 n ( f ) {\displaystyle \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 by real polynomials of no more than degree n, and f(x) = |x| 1913 ? ? ?
Tribonacci constant 1.83928 67552 14161 13255 1 + 19 + 3 33 3 + 19 − 3 33 3 3 = 1 + 4 cosh ⁡ ( 1 3 cosh − 1 ⁡ ( 2 + 3 8 ) ) 3 {\textstyle {\frac {1+{\sqrt{19+3{\sqrt {33}}}}+{\sqrt{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 {\displaystyle x^{3}-x^{2}-x-1=0}

1914 to 1963 ✗ ✓ ✓
Brun's constant B 2 {\displaystyle B_{2}} 1.90216 05831 04 ∑ p ( 1 p + 1 p + 2 ) = ( 1 3 + 1 5 ) + ( 1 5 + 1 7 ) + ( 1 11 + 1 13 ) + ⋯ {\displaystyle \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 ? ? ?
Twin primes constant C 2 {\displaystyle C_{2}} 0.66016 18158 46869 57392 ∏ p p r i m e p ≥ 3 ( 1 − 1 ( p − 1 ) 2 ) {\displaystyle \prod _{\textstyle {p\;{\rm {prime}} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)} 1922 ? ? ?
Plastic ratio ρ {\displaystyle \rho } 1.32471 79572 44746 02596 1 + 1 + 1 + ⋯ 3 3 3 = 1 2 + 69 18 3 + 1 2 − 69 18 3 {\displaystyle {\sqrt{1+\!{\sqrt{1+\!{\sqrt{1+\cdots }}}}}}=\textstyle {\sqrt{{\frac {1}{2}}+{\frac {\sqrt {69}}{18}}}}+{\sqrt{{\frac {1}{2}}-{\frac {\sqrt {69}}{18}}}}}

Real root of x 3 = x + 1 {\displaystyle x^{3}=x+1}

1924 ✗ ✓ ✓
Bloch's constant B {\displaystyle B} 0.4332 ≤ B ≤ 0.4719 {\displaystyle 0.4332\leq B\leq 0.4719} The best known bounds are 3 4 + 2 × 10 − 4 ≤ B ≤ 3 − 1 2 ⋅ Γ ( 1 3 ) Γ ( 11 12 ) Γ ( 1 4 ) {\displaystyle {\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 ? ? ?
Z score for the 97.5 percentile point z .975 {\displaystyle z_{.975}} 1.95996 39845 40054 23552 2 erf − 1 ⁡ ( 0.95 ) {\displaystyle {\sqrt {2}}\operatorname {erf} ^{-1}(0.95)} where erf(x) is the inverse error function

Real number z {\displaystyle z} such that 1 2 π ∫ − ∞ z e − x 2 / 2 d x = 0.975 {\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-x^{2}/2}\,\mathrm {d} x=0.975}

1925 ? ? ?
Landau's constant L {\displaystyle L} 0.5 < L ≤ 0.54326 {\displaystyle 0.5<L\leq 0.54326} The best known bounds are 0.5 < L ≤ Γ ( 1 3 ) Γ ( 5 6 ) Γ ( 1 6 ) {\displaystyle 0.5<L\leq {\frac {\Gamma ({\frac {1}{3}})\Gamma ({\frac {5}{6}})}{\Gamma ({\frac {1}{6}})}}} 1929 ? ? ?
Landau's third constant A {\displaystyle A} 0.5 < A ≤ 0.7853 {\displaystyle 0.5<A\leq 0.7853} 1929 ? ? ?
Prouhet–Thue–Morse constant τ {\displaystyle \tau } 0.41245 40336 40107 59778 ∑ n = 0 ∞ t n 2 n + 1 = 1 4 [ 2 − ∏ n = 0 ∞ ( 1 − 1 2 2 n ) ] {\displaystyle \sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}={\frac {1}{4}}\left}

where t n {\displaystyle {t_{n}}} is the n term of the Thue–Morse sequence

1929 ✗ ✗ ?
Golomb–Dickman constant λ {\displaystyle \lambda } 0.62432 99885 43550 87099 ∫ 0 1 e L i ( t ) d t = ∫ 0 ∞ ρ ( t ) t + 2 d t {\displaystyle \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 c {\displaystyle c} 0.98943 12738 31146 95174 lim n → ∞ ( L n − 4 π 2 ln ⁡ ( 2 n + 1 ) ) = 4 π 2 ( − Γ ′ ( 1 2 ) Γ ( 1 2 ) + ∑ k = 1 ∞ 2 ln ⁡ k 4 k 2 − 1 ) {\displaystyle \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 ? ? ?
Feller–Tornier constant C F T {\displaystyle {\mathcal {C}}_{\mathrm {FT} }} 0.66131 70494 69622 33528 1 2 ∏ p  prime ( 1 − 2 p 2 ) + 1 2 = 3 π 2 ∏ p  prime ( 1 − 1 p 2 − 1 ) + 1 2 {\displaystyle {{\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 {\displaystyle 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 ✗ ✗ ?
Salem constant σ 10 {\displaystyle \sigma _{10}} 1.17628 08182 59917 50654 Largest real root of x 10 + x 9 − x 7 − x 6 − x 5 − x 4 − x 3 + x + 1 = 0 {\displaystyle x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1=0} 1933 ✗ ✓ ✓
Khinchin's constant K 0 {\displaystyle K_{0}} 2.68545 20010 65306 44530  ∏ n = 1 ∞ [ 1 + 1 n ( n + 2 ) ] log 2 ⁡ ( n ) {\displaystyle \prod _{n=1}^{\infty }\left^{\log _{2}(n)}} 1934 ? ? ?
Lévy's constant (1) β {\displaystyle \beta } 1.18656 91104 15625 45282 π 2 12 ln ⁡ 2 {\displaystyle {\frac {\pi ^{2}}{12\,\ln 2}}} 1935 ? ? ?
Lévy's constant (2) e β {\displaystyle e^{\beta }} 3.27582 29187 21811 15978 e π 2 / ( 12 ln ⁡ 2 ) {\displaystyle e^{\pi ^{2}/(12\ln 2)}} 1936 ? ? ?
Copeland–Erdős constant C C E {\displaystyle {\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 ...

1946 ✗ ? ?
Mills' constant A {\displaystyle A} 1.30637 78838 63080 69046 Smallest positive real number A such that ⌊ A 3 n ⌋ {\displaystyle \lfloor A^{3^{n}}\rfloor } is prime for all positive integers n 1947 ? ? ?
Gompertz constant δ {\displaystyle \delta } 0.59634 73623 23194 07434 ∫ 0 ∞ e − x 1 + x d x = ∫ 0 1 d x 1 − ln ⁡ x = 1 1 + 1 1 + 1 1 + 2 1 + 2 1 + 3 1 + 3 / ⋯ {\displaystyle \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 ? ? ?
de Bruijn–Newman constant Λ {\displaystyle \Lambda } 0 ≤ Λ ≤ 0.2 {\displaystyle 0\leq \Lambda \leq 0.2} The number Λ such that H ( λ , z ) = ∫ 0 ∞ e λ u 2 Φ ( u ) cos ⁡ ( z u ) d u {\displaystyle 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 Φ ( u ) = ∑ n = 1 ∞ ( 2 π 2 n 4 e 9 u − 3 π n 2 e 5 u ) e − π n 2 e 4 u {\displaystyle \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 π ln ⁡ 2 {\displaystyle {\frac {\pi }{\ln 2}}} 4.53236 01418 27193 80962 π ln ⁡ 2 {\displaystyle {\frac {\pi }{\ln 2}}} Before 1958 ✗ ? ?
Magic angle θ m {\displaystyle \theta _{\mathrm {m} }} 0.95531 66181 245092 78163 arctan ⁡ 2 = arccos ⁡ 1 3 ≈ 54.7356 ∘ {\displaystyle \arctan {\sqrt {2}}=\arccos {\tfrac {1}{\sqrt {3}}}\approx \textstyle {54.7356}^{\circ }} Before 1959 ✗ ✗ ✓
Artin's constant C A r t i n {\displaystyle C_{\mathrm {Artin} }} 0.37395 58136 19202 28805 ∏ p  prime ( 1 − 1 p ( p − 1 ) ) {\displaystyle \prod _{p{\text{ prime}}}\left(1-{\frac {1}{p(p-1)}}\right)} Before 1961 ? ? ?
Porter's constant C {\displaystyle C} 1.46707 80794 33975 47289 6 ln ⁡ 2 π 2 ( 3 ln ⁡ 2 + 4 γ − 24 π 2 ζ ′ ( 2 ) − 2 ) − 1 2 {\displaystyle {\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 ? ? ?
Lochs constant L {\displaystyle L} 0.97027 01143 92033 92574 6 ln ⁡ 2 ln ⁡ 10 π 2 {\displaystyle {\frac {6\ln 2\ln 10}{\pi ^{2}}}} 1964 ? ? ?
DeVicci's tesseract constant 1.00743 47568 84279 37609 The largest cube that can pass through a 4D hypercube.

Positive root of 4 x 8 − 28 x 6 − 7 x 4 + 16 x 2 + 16 = 0 {\displaystyle 4x^{8}{-}28x^{6}{-}7x^{4}{+}16x^{2}{+}16=0}

1966 ✗ ✓ ✓
Lieb's square ice constant 1.53960 07178 39002 03869 ( 4 3 ) 3 2 = 8 3 3 {\displaystyle \left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}} 1967 ✗ ✓ ✓
Niven's constant C {\displaystyle C} 1.70521 11401 05367 76428 1 + ∑ n = 2 ∞ ( 1 − 1 ζ ( n ) ) {\displaystyle 1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)}}\right)} 1969 ? ? ?
Stephens' constant 0.57595 99688 92945 43964 ∏ p  prime ( 1 − p p 3 − 1 ) {\displaystyle \prod _{p{\text{ prime}}}\left(1-{\frac {p}{p^{3}-1}}\right)} 1969 ? ? ?
Regular paperfolding sequence P {\displaystyle P} 0.85073 61882 01867 26036 ∑ n = 0 ∞ 8 2 n 2 2 n + 2 − 1 = ∑ n = 0 ∞ 1 2 2 n 1 − 1 2 2 n + 2 {\displaystyle \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 ✗ ✗ ?
Reciprocal Fibonacci constant ψ {\displaystyle \psi } 3.35988 56662 43177 55317 ∑ n = 1 ∞ 1 F n = 1 1 + 1 1 + 1 2 + 1 3 + 1 5 + 1 8 + 1 13 + ⋯ {\displaystyle \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 n Fibonacci number

1974 ✗ ? ?
Chvátal–Sankoff constant for the binary alphabet γ 2 {\displaystyle \gamma _{2}} 0.788071 ≤ γ 2 ≤ 0.826280 {\displaystyle 0.788071\leq \gamma _{2}\leq 0.826280} lim n → ∞ E ⁡ [ λ n , 2 ] n {\displaystyle \lim _{n\to \infty }{\frac {\operatorname {E} }{n}}}

where E is the expected longest common subsequence of two random length-n binary strings

1975 ? ? ?
Feigenbaum constant δ δ {\displaystyle \delta } 4.66920 16091 02990 67185 lim n → ∞ a n + 1 − a n a n + 2 − a n + 1 {\displaystyle \lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{a_{n+2}-a_{n+1}}}}

where the sequence an is given by n-th period-doubling bifurcation of logistic map x k + 1 = a x k ( 1 − x k ) {\displaystyle x_{k+1}=ax_{k}(1-x_{k})} or any other one-dimensional map with a single quadratic maximum

1975 ? ? ?
Chaitin's constants Ω {\displaystyle \Omega } In general they are uncomputable numbers.
But one such number is 0.00787 49969 97812 3844.
 ∑ p ∈ P 2 − | p | {\displaystyle \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 ✗ ✗ ✗
Robbins constant Δ ( 3 ) {\displaystyle \Delta (3)} 0.66170 71822 67176 23515 4 + 17 2 − 6 3 − 7 π 105 + ln ⁡ ( 1 + 2 ) 5 + 2 ln ⁡ ( 2 + 3 ) 5 {\displaystyle {\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 ✗ ✗ ✓
Weierstrass constant 0.47494 93799 87920 65033 2 5 / 4 π e π / 8 Γ ( 1 4 ) 2 {\displaystyle {\frac {2^{5/4}{\sqrt {\pi }}\,e^{\pi /8}}{\Gamma ({\frac {1}{4}})^{2}}}} Before 1978 ✗ ✗ ?
Fransén–Robinson constant F {\displaystyle F} 2.80777 02420 28519 36522 ∫ 0 ∞ d x Γ ( x ) = e + ∫ 0 ∞ e − x π 2 + ln 2 ⁡ x d x {\displaystyle \int _{0}^{\infty }{\frac {dx}{\Gamma (x)}}=e+\int _{0}^{\infty }{\frac {e^{-x}}{\pi ^{2}+\ln ^{2}x}}\,dx} 1978 ? ? ?
Feigenbaum constant α α {\displaystyle \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 ? ? ?
Second du Bois-Reymond constant C 2 {\displaystyle C_{2}} 0.19452 80494 65325 11361 e 2 − 7 2 = ∫ 0 ∞ | d d t ( sin ⁡ t t ) 2 | d t − 1 {\displaystyle {\frac {e^{2}-7}{2}}=\int _{0}^{\infty }\left|{{\frac {d}{dt}}\left({\frac {\sin t}{t}}\right)^{2}}\right|\,dt-1} 1983 ✗ ✗ ?
Erdős–Tenenbaum–Ford constant δ {\displaystyle \delta } 0.08607 13320 55934 20688 1 − 1 + log ⁡ log ⁡ 2 log ⁡ 2 {\displaystyle 1-{\frac {1+\log \log 2}{\log 2}}} 1984 ? ? ?
Conway's constant λ {\displaystyle \lambda } 1.30357 72690 34296 39125 Real root of the polynomial:

x 71 − x 69 − 2 x 68 − x 67 + 2 x 66 + 2 x 65 + x 64 − x 63 − x 62 − x 61 − x 60 − x 59 + 2 x 58 + 5 x 57 + 3 x 56 − 2 x 55 − 10 x 54 − 3 x 53 − 2 x 52 + 6 x 51 + 6 x 50 + x 49 + 9 x 48 − 3 x 47 − 7 x 46 − 8 x 45 − 8 x 44 + 10 x 43 + 6 x 42 + 8 x 41 − 5 x 40 − 12 x 39 + 7 x 38 − 7 x 37 + 7 x 36 + x 35 − 3 x 34 + 10 x 33 + x 32 − 6 x 31 − 2 x 30 − 10 x 29 − 3 x 28 + 2 x 27 + 9 x 26 − 3 x 25 + 14 x 24 − 8 x 23 − 7 x 21 + 9 x 20 + 3 x 19 − 4 x 18 − 10 x 17 − 7 x 16 + 12 x 15 + 7 x 14 + 2 x 13 − 12 x 12 − 4 x 11 − 2 x 10 + 5 x 9 + x 7 − 7 x 6 + 7 x 5 − 4 x 4 + 12 x 3 − 6 x 2 + 3 x − 6   =   0 {\displaystyle {\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 ✗ ✓ ✓
Hafner–Sarnak–McCurley constant σ {\displaystyle \sigma } 0.35323 63718 54995 98454 ∏ p  prime ( 1 − ( 1 − ∏ n ≥ 1 ( 1 − 1 p n ) ) 2 ) {\displaystyle \prod _{p{\text{ prime}}}{\left(1-\left(1-\prod _{n\geq 1}\left(1-{\frac {1}{p^{n}}}\right)\right)^{2}\right)}\!} 1991 ? ? ?
Backhouse's constant B {\displaystyle B} 1.45607 49485 82689 67139 lim k → ∞ | q k + 1 q k | where: Q ( x ) = 1 P ( x ) = ∑ k = 1 ∞ q k x k {\displaystyle \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 + ∑ k = 1 ∞ p k x k = 1 + 2 x + 3 x 2 + 5 x 3 + ⋯ {\displaystyle P(x)=1+\sum _{k=1}^{\infty }{p_{k}x^{k}}=1+2x+3x^{2}+5x^{3}+\cdots } where pk is the k prime number

1995 ? ? ?
Viswanath constant 1.13198 82487 943 lim n → ∞ | f n | 1 n {\displaystyle \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 ? ? ?
Komornik–Loreti constant q {\displaystyle q} 1.78723 16501 82965 93301 Real number q {\displaystyle q} such that 1 = ∑ k = 1 ∞ t k q k {\displaystyle 1=\sum _{k=1}^{\infty }{\frac {t_{k}}{q^{k}}}} , or ∏ n = 0 ∞ ( 1 − 1 q 2 n ) + q − 2 q − 1 = 0 {\displaystyle \prod _{n=0}^{\infty }\left(1-{\frac {1}{q^{2^{n}}}}\right)+{\frac {q-2}{q-1}}=0}

where tk is the k term of the Thue–Morse sequence

1998 ✗ ✗ ?
Embree–Trefethen constant β ⋆ {\displaystyle \beta ^{\star }} 0.70258 1999 ? ? ?
Heath-Brown–Moroz constant C {\displaystyle C} 0.00131 76411 54853 17810 ∏ p  prime ( 1 − 1 p ) 7 ( 1 + 7 p + 1 p 2 ) {\displaystyle \prod _{p{\text{ prime}}}\left(1-{\frac {1}{p}}\right)^{7}\left(1+{\frac {7p+1}{p^{2}}}\right)} 1999 ? ? ?
MRB constant S {\displaystyle S} 0.18785 96424 62067 12024 ∑ n = 1 ∞ ( − 1 ) n ( n 1 / n − 1 ) = − 1 1 + 2 2 − 3 3 + ⋯ {\displaystyle \sum _{n=1}^{\infty }(-1)^{n}(n^{1/n}-1)=-{\sqrt{1}}+{\sqrt{2}}-{\sqrt{3}}+\cdots } 1999 ? ? ?
Prime constant ρ {\displaystyle \rho } 0.41468 25098 51111 66024 ∑ p  prime 1 2 p = 1 4 + 1 8 + 1 32 + ⋯ {\displaystyle \sum _{p{\text{ prime}}}{\frac {1}{2^{p}}}={\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{32}}+\cdots } 1999 ✗ ? ?
Somos' quadratic recurrence constant σ {\displaystyle \sigma } 1.66168 79496 33594 12129 ∏ n = 1 ∞ n 1 / 2 n = 1 2 3 ⋯ = 1 1 / 2 2 1 / 4 3 1 / 8 ⋯ {\displaystyle \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 {\displaystyle \alpha } 1.18745 23511 26501 05459 x n + 1 = ( 1 + 1 x n ) n  for  n = 1 , 2 , 3 , … {\displaystyle 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 ? ? ?
Logarithmic capacity of the unit disk 0.59017 02995 08048 11302 Γ ( 1 4 ) 2 4 π 3 / 2 = ϖ π 2 {\displaystyle {\frac {\Gamma ({\tfrac {1}{4}})^{2}}{4\pi ^{3/2}}}={\frac {\varpi }{\pi {\sqrt {2}}}}} where ϖ {\displaystyle \varpi } is the lemniscate constant. Before 2003 ✗ ✗ ?
Taniguchi constant 0.67823 44919 17391 97803 ∏ p  prime ( 1 − 3 p 3 + 2 p 4 + 1 p 5 − 1 p 6 ) {\displaystyle \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 ? ? ?

Mathematical constants sorted by their representations as continued fractions

The following list includes the continued fractions of some constants and is sorted by their representations. Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one. Decimal representations are rounded or padded to 10 places if the values are known.

Name Symbol Set Decimal expansion Continued fraction Notes
Zero 0 Z {\displaystyle \mathbb {Z} } 0.00000 00000
Golomb–Dickman constant λ {\displaystyle \lambda } 0.62432 99885 E. Weisstein noted that the continued fraction has an unusually large number of 1s.
Cahen's constant C 2 {\displaystyle C_{2}} R ∖ A {\displaystyle \mathbb {R} \setminus \mathbb {A} } 0.64341 05463 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 γ {\displaystyle \gamma } 0.57721 56649 Using the continued fraction expansion, it was shown that if γ is rational, its denominator must exceed 10.
First continued fraction constant C 1 {\displaystyle C_{1}} R ∖ A {\displaystyle \mathbb {R} \setminus \mathbb {A} } 0.69777 46579 Equal to the ratio I 1 ( 2 ) / I 0 ( 2 ) {\displaystyle I_{1}(2)/I_{0}(2)} of modified Bessel functions of the first kind evaluated at 2.
Catalan's constant G {\displaystyle G} 0.91596 55942 Computed up to 4851389025 terms by E. Weisstein.
One half 1/2 Q {\displaystyle \mathbb {Q} } 0.50000 00000
Prouhet–Thue–Morse constant τ {\displaystyle \tau } R ∖ A {\displaystyle \mathbb {R} \setminus \mathbb {A} } 0.41245 40336 Infinitely many partial quotients are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.
Copeland–Erdős constant C C E {\displaystyle {\mathcal {C}}_{CE}} R ∖ Q {\displaystyle \mathbb {R} \setminus \mathbb {Q} } 0.23571 11317 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.
Base 10 Champernowne constant C 10 {\displaystyle C_{10}} R ∖ A {\displaystyle \mathbb {R} \setminus \mathbb {A} } 0.12345 67891 Champernowne constants in any base exhibit sporadic large numbers; the 40th term in C 10 {\displaystyle C_{10}} has 2504 digits.
One 1 N {\displaystyle \mathbb {N} } 1.00000 00000
Phi, Golden ratio φ {\displaystyle \varphi } A {\displaystyle \mathbb {A} } 1.61803 39887 The convergents are ratios of successive Fibonacci numbers.
Brun's constant B 2 {\displaystyle B_{2}} 1.90216 05831 The n roots of the denominators of the n convergents are close to Khinchin's constant, suggesting that B 2 {\displaystyle B_{2}} is irrational. If true, this will prove the twin prime conjecture.
Square root of 2 2 {\displaystyle {\sqrt {2}}} A {\displaystyle \mathbb {A} } 1.41421 35624 The convergents are ratios of successive Pell numbers.
Two 2 N {\displaystyle \mathbb {N} } 2.00000 00000
Euler's number e {\displaystyle e} R ∖ A {\displaystyle \mathbb {R} \setminus \mathbb {A} } 2.71828 18285 The continued fraction expansion has the pattern .
Khinchin's constant K 0 {\displaystyle K_{0}} 2.68545 20011 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 N {\displaystyle \mathbb {N} } 3.00000 00000
Pi π {\displaystyle \pi } R ∖ A {\displaystyle \mathbb {R} \setminus \mathbb {A} } 3.14159 26536 The first few convergents (3, 22/7, 333/106, 355/113, ...) are among the best-known and most widely used historical approximations of π.

Sequences of constants

Name Symbol Formula Year Set
Harmonic number H n {\displaystyle H_{n}} ∑ k = 1 n 1 k {\displaystyle \sum _{k=1}^{n}{\frac {1}{k}}} Antiquity Q {\displaystyle \mathbb {Q} }
Gregory coefficients G n {\displaystyle G_{n}} 1 n ! ∫ 0 1 x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) d x = ∫ 0 1 ( x n ) d x {\displaystyle {\frac {1}{n!}}\int _{0}^{1}x(x-1)(x-2)\cdots (x-n+1)\,dx=\int _{0}^{1}{\binom {x}{n}}\,dx} 1670 Q {\displaystyle \mathbb {Q} }
Bernoulli number B n ± {\displaystyle B_{n}^{\pm }} t 2 ( coth ⁡ t 2 ± 1 ) = ∑ m = 0 ∞ B m ± t m m ! {\displaystyle {\frac {t}{2}}\left(\operatorname {coth} {\frac {t}{2}}\pm 1\right)=\sum _{m=0}^{\infty }{\frac {B_{m}^{\pm {}}t^{m}}{m!}}} 1689 Q {\displaystyle \mathbb {Q} }
Hermite constants γ n {\displaystyle \gamma _{n}} For a lattice L in Euclidean space R with unit covolume, i.e. vol(R/L) = 1, let λ1(L) denote the least length of a nonzero element of L. Then √γnn is the maximum of λ1(L) over all such lattices L. 1822 to 1901 R {\displaystyle \mathbb {R} }
Hafner–Sarnak–McCurley constant D ( n ) {\displaystyle D(n)} D ( n ) = ∏ k = 1 ∞ { 1 − [ 1 − ∏ j = 1 n ( 1 − p k − j ) ] 2 } {\displaystyle D(n)=\prod _{k=1}^{\infty }\left\{1-\left^{2}\right\}} 1883 R {\displaystyle \mathbb {R} }
Stieltjes constants γ n {\displaystyle \gamma _{n}} ( − 1 ) n n ! 2 π ∫ 0 2 π e − n i x ζ ( e i x + 1 ) d x . {\displaystyle {\frac {(-1)^{n}n!}{2\pi }}\int _{0}^{2\pi }e^{-nix}\zeta \left(e^{ix}+1\right)dx.} before 1894 R {\displaystyle \mathbb {R} }
Favard constants K r {\displaystyle K_{r}} 4 π ∑ n = 0 ∞ ( ( − 1 ) n 2 n + 1 ) r + 1 = 4 π ( ( − 1 ) 0 ( r + 1 ) 1 r + ( − 1 ) 1 ( r + 1 ) 3 r + ( − 1 ) 2 ( r + 1 ) 5 r + ( − 1 ) 3 ( r + 1 ) 7 r + ⋯ ) {\displaystyle {\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 R {\displaystyle \mathbb {R} }
Generalized Brun's Constant B n {\displaystyle B_{n}} ∑ p ( 1 p + 1 p + n ) {\displaystyle {\sum \limits _{p}({\frac {1}{p}}+{\frac {1}{p+n}})}} where the sum ranges over all primes p such that p + n is also a prime 1919 R {\displaystyle \mathbb {R} }
Champernowne constants C b {\displaystyle C_{b}} Defined by concatenating representations of successive integers in base b.

C b = ∑ n = 1 ∞ n b ( ∑ k = 1 n ⌈ log b ⁡ ( k + 1 ) ⌉ ) {\displaystyle C_{b}=\sum _{n=1}^{\infty }{\frac {n}{b^{\left(\sum _{k=1}^{n}\lceil \log _{b}(k+1)\rceil \right)}}}}

1933 R ∖ A {\displaystyle \mathbb {R} \setminus \mathbb {A} }
Lagrange number L ( n ) {\displaystyle L(n)} 9 − 4 m n 2 {\displaystyle {\sqrt {9-{\frac {4}{{m_{n}}^{2}}}}}} where m n {\displaystyle m_{n}} is the nth smallest number such that m 2 + x 2 + y 2 = 3 m x y {\displaystyle m^{2}+x^{2}+y^{2}=3mxy\,} has positive (x,y). before 1957 A {\displaystyle \mathbb {A} }
Feller's coin-tossing constants α k , β k {\displaystyle \alpha _{k},\beta _{k}} α k {\displaystyle \alpha _{k}} is the smallest positive real root of x k + 1 = 2 k + 1 ( x − 1 ) , β k = 2 − α k k + 1 − k α k {\displaystyle x^{k+1}=2^{k+1}(x-1),\beta _{k}={\frac {2-\alpha _{k}}{k+1-k\alpha _{k}}}} 1968 A {\displaystyle \mathbb {A} }
Stoneham number α b , c {\displaystyle \alpha _{b,c}} ∑ n = c k > 1 1 b n n = ∑ k = 1 ∞ 1 b c k c k {\displaystyle \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 R ∖ A {\displaystyle \mathbb {R} \setminus \mathbb {A} }
Beraha constants B ( n ) {\displaystyle B(n)} 2 + 2 cos ⁡ ( 2 π n ) {\displaystyle 2+2\cos \left({\frac {2\pi }{n}}\right)} 1974 A {\displaystyle \mathbb {A} }
Chvátal–Sankoff constants γ k {\displaystyle \gamma _{k}} lim n → ∞ E [ λ n , k ] n {\displaystyle \lim _{n\to \infty }{\frac {E}{n}}} 1975 R {\displaystyle \mathbb {R} }
Hyperharmonic number H n ( r ) {\displaystyle H_{n}^{(r)}} ∑ k = 1 n H k ( r − 1 ) {\displaystyle \sum _{k=1}^{n}H_{k}^{(r-1)}} and H n ( 0 ) = 1 n {\displaystyle H_{n}^{(0)}={\frac {1}{n}}} 1995 Q {\displaystyle \mathbb {Q} }
Gregory number G x {\displaystyle G_{x}} ∑ n = 0 ∞ ( − 1 ) n 1 ( 2 n + 1 ) x 2 n + 1 = arccot ⁡ ( x ) {\displaystyle \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 R ∖ A {\displaystyle \mathbb {R} \setminus \mathbb {A} }
Metallic mean n + n 2 + 4 2 {\displaystyle {\frac {n+{\sqrt {n^{2}+4}}}{2}}} before 1998 A {\displaystyle \mathbb {A} }

See also

Notes

  1. Both i and −i are roots of this equation, though neither root is truly "positive" nor more fundamental than the other as they are algebraically equivalent. The distinction between signs of i and −i is in some ways arbitrary, but a useful notational device. See imaginary unit for more information.

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  60. "Engineering Statistics Handbook: Confidence Limits for the Mean". National Institute of Standards and Technology. Archived from the original on 5 February 2008. Retrieved 4 February 2008. Although the choice of confidence coefficient is somewhat arbitrary, in practice 90%, 95%, and 99% intervals are often used, with 95% being the most commonly used.
  61. Olson, Eric T; Olson, Tammy Perry (2000), Real-Life Math: Statistics, Walch Publishing, p. 66, ISBN 0-8251-3863-9, While other stricter, or looser, limits may be chosen, the 95 percent interval is very often preferred by statisticians.
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Site MathWorld Wolfram.com

  1. Weisstein, Eric W. "Pi Formulas". MathWorld.
  2. Weisstein, Eric W. "Pythagoras's Constant". MathWorld.
  3. Weisstein, Eric W. "Theodorus's Constant". MathWorld.
  4. Weisstein, Eric W. "Golden Ratio". MathWorld.
  5. Weisstein, Eric W. "Silver Ratio". MathWorld.
  6. Weisstein, Eric W. "Delian Constant". MathWorld.
  7. Weisstein, Eric W. "Self-Avoiding Walk Connective Constant". MathWorld.
  8. Weisstein, Eric W. "Polygon Inscribing". MathWorld.
  9. Weisstein, Eric W. "Wallis's Constant". MathWorld.
  10. Weisstein, Eric W. "e". MathWorld.
  11. Weisstein, Eric W. "Natural Logarithm of 2". MathWorld.
  12. Weisstein, Eric W. "Lemniscate Constant". MathWorld.
  13. Weisstein, Eric W. "Euler–Mascheroni Constant". MathWorld.
  14. Weisstein, Eric W. "Erdos-Borwein Constant". MathWorld.
  15. Weisstein, Eric W. "Omega Constant". MathWorld.
  16. Weisstein, Eric W. "Apéry's Constant". MathWorld.
  17. Weisstein, Eric W. "Laplace Limit". MathWorld.
  18. Weisstein, Eric W. "Soldner's Constant". MathWorld.
  19. Weisstein, Eric W. "Gauss's Constant". MathWorld.
  20. Weisstein, Eric W. "Hermite Constants". MathWorld.
  21. Weisstein, Eric W. "Liouville's Constant". MathWorld.
  22. Weisstein, Eric W. "Continued Fraction Constants". MathWorld.
  23. Weisstein, Eric W. "Ramanujan Constant". MathWorld.
  24. Weisstein, Eric W. "Glaisher-Kinkelin Constant". MathWorld.
  25. Weisstein, Eric W. "Catalan's Constant". MathWorld.
  26. ^ Weisstein, Eric W. "Dottie Number". MathWorld.
  27. Weisstein, Eric W. "Mertens Constant". MathWorld.
  28. Weisstein, Eric W. "Universal Parabolic Constant". MathWorld.
  29. Weisstein, Eric W. "Cahen's Constant". MathWorld.
  30. Weisstein, Eric W. "Gelfonds Constant". MathWorld.
  31. Weisstein, Eric W. "Gelfond-Schneider Constant". MathWorld.
  32. Weisstein, Eric W. "Favard Constants". MathWorld.
  33. Weisstein, Eric W. "Golden Angle". MathWorld.
  34. Weisstein, Eric W. "Sierpinski Constant". MathWorld.
  35. Weisstein, Eric W. "Landau-Ramanujan Constant". MathWorld.
  36. Weisstein, Eric W. "Nielsen-Ramanujan Constants". MathWorld.
  37. Weisstein, Eric W. "Gieseking's Constant". MathWorld.
  38. Weisstein, Eric W. "Bernstein's Constant". MathWorld.
  39. Weisstein, Eric W. "Tribonacci Constant". MathWorld.
  40. Weisstein, Eric W. "Brun's Constant". MathWorld.
  41. Weisstein, Eric W. "Twin Primes Constant". MathWorld.
  42. Weisstein, Eric W. "Plastic Constant". MathWorld.
  43. Weisstein, Eric W. "Bloch Constant". MathWorld.
  44. Weisstein, Eric W. "Confidence Interval". MathWorld.
  45. Weisstein, Eric W. "Landau Constant". MathWorld.
  46. Weisstein, Eric W. "Thue-Morse Constant". MathWorld.
  47. Weisstein, Eric W. "Golomb–Dickman Constant". MathWorld.
  48. ^ Weisstein, Eric W. "Lebesgue Constants". MathWorld.
  49. Weisstein, Eric W. "Feller–Tornier Constant". MathWorld.
  50. Weisstein, Eric W. "Champernowne Constant". MathWorld.
  51. Weisstein, Eric W. "Salem Constants". MathWorld.
  52. Weisstein, Eric W. "Khinchin's Constant". MathWorld.
  53. Weisstein, Eric W. "Levy Constant". MathWorld.
  54. Weisstein, Eric W. "Levy Constant". MathWorld.
  55. Weisstein, Eric W. "Copeland–Erdos Constant". MathWorld.
  56. Weisstein, Eric W. "Mills Constant". MathWorld.
  57. Weisstein, Eric W. "Gompertz Constant". MathWorld.
  58. Weisstein, Eric W. "Artin's Constant". MathWorld.
  59. Weisstein, Eric W. "Porter's Constant". MathWorld.
  60. Weisstein, Eric W. "Lochs' Constant". MathWorld.
  61. Weisstein, Eric W. "Liebs Square Ice Constant". MathWorld.
  62. Weisstein, Eric W. "Niven's Constant". MathWorld.
  63. Weisstein, Eric W. "Stephen's Constant". MathWorld.
  64. Weisstein, Eric W. "Paper Folding Constant". MathWorld.
  65. Weisstein, Eric W. "Reciprocal Fibonacci Constant". MathWorld.
  66. ^ Weisstein, Eric W. "Feigenbaum Constant". MathWorld.
  67. Weisstein, Eric W. "Chaitin's Constant". MathWorld.
  68. Weisstein, Eric W. "Robbins Constant". MathWorld.
  69. Weisstein, Eric W. "Weierstrass Constant". MathWorld.
  70. Weisstein, Eric W. "Fransen-Robinson Constant". MathWorld.
  71. Weisstein, Eric W. "du Bois-Reymond Constants". MathWorld.
  72. Weisstein, Eric W. "Conway's Constant". MathWorld.
  73. Weisstein, Eric W. "Hafner-Sarnak-McCurley Constant". MathWorld.
  74. Weisstein, Eric W. "Backhouse's Constant". MathWorld.
  75. Weisstein, Eric W. "Random Fibonacci Sequence". MathWorld.
  76. Weisstein, Eric W. "Komornik-Loreti Constant". MathWorld.
  77. Weisstein, Eric W. "Heath-Brown-Moroz Constant". MathWorld.
  78. Weisstein, Eric W. "MRB Constant". MathWorld.
  79. ^ Weisstein, Eric W. "Somos's Quadratic Recurrence Constant". MathWorld.
  80. Weisstein, Eric W. "Foias Constant". MathWorld.
  81. Weisstein, Eric W. "Logarithmic Capacity". MathWorld.
  82. Weisstein, Eric W. "Taniguchis Constant". MathWorld.
  83. Weisstein, Eric W. "Golomb-Dickman Constant Continued Fraction". MathWorld.
  84. Weisstein, Eric W. "Catalan's Constant Continued Fraction". MathWorld.
  85. Weisstein, Eric W. "Copeland–Erdős Constant Continued Fraction". MathWorld.
  86. "Hermite Constants".
  87. Weisstein, Eric W. "Relatively Prime". MathWorld.
  88. "Favard Constants".

Site OEIS.org

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  2. OEIS: A019692
  3. OEIS: A002193
  4. OEIS: A002194
  5. OEIS: A002163
  6. OEIS: A001622
  7. OEIS: A014176
  8. OEIS: A002580
  9. OEIS: A002581
  10. OEIS: A010774
  11. OEIS: A092526
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  14. OEIS: A007493
  15. OEIS: A001113
  16. OEIS: A002162
  17. OEIS: A062539
  18. OEIS: A001620
  19. OEIS: A065442
  20. OEIS: A030178
  21. ^ OEIS: A002117
  22. OEIS: A033259
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  27. OEIS: A052119
  28. OEIS: A060295
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  31. OEIS: A003957
  32. OEIS: A077761
  33. OEIS: A103710
  34. OEIS: A118227
  35. OEIS: A039661
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  37. OEIS: A111003
  38. OEIS: A131988
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  41. OEIS: A072691
  42. OEIS: A143298
  43. OEIS: A073001
  44. OEIS: A058265
  45. ^ OEIS: A065421
  46. OEIS: A005597
  47. ^ OEIS: A060006
  48. ^ OEIS: A085508
  49. OEIS: A220510
  50. OEIS: A081760
  51. ^ OEIS: A014571
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  53. OEIS: A243277
  54. OEIS: A065493
  55. OEIS: A033307
  56. ^ OEIS: A073011
  57. OEIS: A002210
  58. OEIS: A100199
  59. OEIS: A086702
  60. ^ OEIS: A033308
  61. OEIS: A051021
  62. ^ OEIS: A073003
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  72. ^ OEIS: A065478
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  74. ^ OEIS: A079586
  75. OEIS: A006890
  76. OEIS: A100264
  77. ^ OEIS: A073012
  78. OEIS: A094692
  79. OEIS: A058655
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  82. OEIS: A074738
  83. OEIS: A014715
  84. ^ OEIS: A085849
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  86. OEIS: A078416
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  88. ^ OEIS: A118228
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  96. OEIS: A006280
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  98. OEIS: A014538
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  101. OEIS: A030167
  102. OEIS: A003417
  103. OEIS: A002211
  104. OEIS: A001203

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