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List of mathematical constants
List containing 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

NameSymbolDecimal expansionFormulaYearSet
Q {\displaystyle \mathbb {Q} } A {\displaystyle \mathbb {A} } P {\displaystyle {\mathcal {P}}}
One11Multiplicative identity of C {\displaystyle \mathbb {C} } .Prehistory✓✓✓
Two22Prehistory✓✓✓
One half⁠1/2⁠0.5Prehistory✓✓✓
Pi π {\displaystyle \pi } 3.14159 26535 89793 23846 23Ratio of a circle's circumference to its diameter.1900 to 1600 BCE 4✗✗✓
Tau τ {\displaystyle \tau } 6.28318 53071 79586 4769256Ratio of a circle's circumference to its radius. Equal to 2 π {\displaystyle 2\pi } 1900 to 1600 BCE 7✗✗✓
Square root of 2,

Pythagoras constant8

 2 {\displaystyle {\sqrt {2}}} 1.41421 35623 73095 04880 910Positive root of x 2 = 2 {\displaystyle x^{2}=2} 1800 to 1600 BCE11✗✓✓
Square root of 3,

Theodorus' constant12

 3 {\displaystyle {\sqrt {3}}} 1.73205 08075 68877 29352 1314Positive root of x 2 = 3 {\displaystyle x^{2}=3} 465 to 398 BCE✗✓✓
Square root of 515 5 {\displaystyle {\sqrt {5}}} 2.23606 79774 99789 69640 16Positive root of x 2 = 5 {\displaystyle x^{2}=5} ✗✓✓
Phi, Golden ratio17 φ {\displaystyle \varphi } or ϕ {\displaystyle \phi } 1.61803 39887 49894 84820 1819 1 + 5 2 {\displaystyle {\frac {1+{\sqrt {5}}}{2}}} ~300 BCE✗✓✓
Silver ratio20 δ S {\displaystyle \delta _{S}} 2.41421 35623 73095 04880 2122 2 + 1 {\displaystyle {\sqrt {2}}+1} ~300 BCE✗✓✓
Zero00Additive identity of C {\displaystyle \mathbb {C} } .300 to 100 BCE23✓✓✓
Negative one−1−1300 to 200 BCE✓✓✓
Cube root of 2 2 3 {\displaystyle {\sqrt[{3}]{2}}} 1.25992 10498 94873 16476 2425Real root of x 3 = 2 {\displaystyle x^{3}=2} 46 to 120 CE26✗✓✓
Cube root of 3 3 3 {\displaystyle {\sqrt[{3}]{3}}} 1.44224 95703 07408 38232 27Real root of x 3 = 3 {\displaystyle x^{3}=3} ✗✓✓
Twelfth root of 228 2 12 {\displaystyle {\sqrt[{12}]{2}}} 1.05946 30943 59295 26456 29Real root of x 12 = 2 {\displaystyle x^{12}=2} ✗✓✓
Supergolden ratio30 ψ {\displaystyle \psi } 1.46557 12318 76768 02665 31 1 + 29 + 3 93 2 3 + 29 − 3 93 2 3 3 {\displaystyle {\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 {\displaystyle x^{3}=x^{2}+1}

✗✓✓
Imaginary unit32 i {\displaystyle i} 0 + 1iPrincipal root of x 2 = − 1 {\displaystyle x^{2}=-1} 331501 to 1576✗✓✓
Connective constant for the hexagonal lattice3435 μ {\displaystyle \mu } 1.84775 90650 22573 51225 3637 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 38✗✓✓
Kepler–Bouwkamp constant39 K ′ {\displaystyle K'} 0.11494 20448 53296 20070 4041 ∏ 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 42???
Wallis's constant2.09455 14815 42326 59148 4344 45 − 1929 18 3 + 45 + 1929 18 3 {\displaystyle {\sqrt[{3}]{\frac {45-{\sqrt {1929}}}{18}}}+{\sqrt[{3}]{\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 number45 e {\displaystyle e} 2.71828 18284 59045 23536 4647 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 } 161848✗✗?
Natural logarithm of 249 ln ⁡ 2 {\displaystyle \ln 2} 0.69314 71805 59945 30941 5051Real 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 }

161952 & 1668 53✗✗✓
Lemniscate constant54 ϖ {\displaystyle \varpi } 2.62205 75542 92119 81046 5556 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 5758 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 constant59 E {\displaystyle E} 1.60669 51524 15291 76378 6061 ∑ 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 } 174962✗??
Omega constant Ω {\displaystyle \Omega } 0.56714 32904 09783 87299 6364 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 constant65 ζ ( 3 ) {\displaystyle \zeta (3)} 1.20205 69031 59594 28539 6667 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 68✗?✓
Laplace limit690.66274 34193 49181 58097 7071Real 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 constant7273 μ {\displaystyle \mu } 1.45136 92348 83381 05028 7475 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 76???
Gauss's constant77 G {\displaystyle G} 0.83462 68416 74073 18628 7879 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.

179980✗✗?
Second Hermite constant81 γ 2 {\displaystyle \gamma _{2}} 1.15470 05383 79251 52901 8283 2 3 {\displaystyle {\frac {2}{\sqrt {3}}}} 1822 to 1901✗✓✓
Liouville's constant84 L {\displaystyle L} 0.11000 10000 00000 00000 0001 8586 ∑ 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 8788 C 1 = [ 0 ; 1 , 2 , 3 , 4 , 5 , . . . ] = I 1 ( 2 ) I 0 ( 2 ) {\displaystyle C_{1}=[0;1,2,3,4,5,...]={\frac {I_{1}(2)}{I_{0}(2)}}} , (see Bessel functions). C 1 ∉ A . {\displaystyle C_{1}\notin \mathbb {A} .} 89185590✗✗?
Ramanujan's constant91262 53741 26407 68743 .99999 99999 99250 073 9293 e π 163 {\displaystyle e^{\pi {\sqrt {163}}}} 1859✗✗?
Glaisher–Kinkelin constant A {\displaystyle A} 1.28242 71291 00622 63687 9495 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 96???
Catalan's constant979899 G {\displaystyle G} 0.91596 55941 77219 01505 100101 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 number1020.73908 51332 15160 64165 103104Real root of cos ⁡ x = x {\displaystyle \cos x=x} 1865 105✗✗?
Meissel–Mertens constant106 M {\displaystyle M} 0.26149 72128 47642 78375 107108 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 constant109 P {\displaystyle P} 2.29558 71493 92638 07403 110111 ln ⁡ ( 1 + 2 ) + 2 = arsinh ⁡ ( 1 ) + 2 {\displaystyle \ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arsinh} (1)+{\sqrt {2}}} Before 1891112✗✗✓
Cahen's constant113 C {\displaystyle C} 0.64341 05462 88338 02618 114115 ∑ 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 constant116 e π {\displaystyle e^{\pi }} 23.14069 26327 79269 0057 117118 ( − 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 } 1900119✗✗?
Gelfond–Schneider constant120 2 2 {\displaystyle 2^{\sqrt {2}}} 2.66514 41426 90225 18865 121122 2 2 {\displaystyle 2^{\sqrt {2}}} Before 1902 123✗✗?
Second Favard constant124 K 2 {\displaystyle K_{2}} 1.23370 05501 36169 82735 125126 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 angle127 g {\displaystyle g} 2.39996 32297 28653 32223 128129 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 constant130 K {\displaystyle K} 2.58498 17595 79253 21706 131132 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 constant133 K {\displaystyle K} 0.76422 36535 89220 66299 134135 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 136???
First Nielsen–Ramanujan constant137 a 1 {\displaystyle a_{1}} 0.82246 70334 24113 21823 138139 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 constant140 G {\displaystyle G} 1.01494 16064 09653 62502 141142 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 constant143 β {\displaystyle \beta } 0.28016 94990 23869 13303 144145 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 [−1, 1] by real polynomials of no more than degree n, and f(x) = |x|1913???
Tribonacci constant1461.83928 67552 14161 13255 147148 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[{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 {\displaystyle x^{3}-x^{2}-x-1=0}

1914 to 1963✗✓✓
Brun's constant149 B 2 {\displaystyle B_{2}} 1.90216 05831 04 150151 ∑ 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 152???
Twin primes constant C 2 {\displaystyle C_{2}} 0.66016 18158 46869 57392 153154 ∏ 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 ratio155 ρ {\displaystyle \rho } 1.32471 79572 44746 02596 156157 1 + 1 + 1 + ⋯ 3 3 3 = 1 2 + 69 18 3 + 1 2 − 69 18 3 {\displaystyle {\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 {\displaystyle x^{3}=x+1}

1924 158✗✓✓
Bloch's constant159 B {\displaystyle B} 0.4332 ≤ B ≤ 0.4719 {\displaystyle 0.4332\leq B\leq 0.4719} 160161The 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 162???
Z score for the 97.5 percentile point163164165166 z .975 {\displaystyle z_{.975}} 1.95996 39845 40054 23552 167168 2 erf − 1 ⁡ ( 0.95 ) {\displaystyle {\sqrt {2}}\operatorname {erf} ^{-1}(0.95)} where erf−1(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 constant169 L {\displaystyle L} 0.5 < L ≤ 0.54326 {\displaystyle 0.5<L\leq 0.54326} 170171The 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 constant172 A {\displaystyle A} 0.5 < A ≤ 0.7853 {\displaystyle 0.5<A\leq 0.7853} 1929???
Prouhet–Thue–Morse constant173 τ {\displaystyle \tau } 0.41245 40336 40107 59778 174175 ∑ 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[2-\prod _{n=0}^{\infty }\left(1-{\frac {1}{2^{2^{n}}}}\right)\right]}

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

1929 176✗✗?
Golomb–Dickman constant177 λ {\displaystyle \lambda } 0.62432 99885 43550 87099 178179 ∫ 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 constants180 c {\displaystyle c} 0.98943 12738 31146 95174 181182 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 183???
Feller–Tornier constant184 C F T {\displaystyle {\mathcal {C}}_{\mathrm {FT} }} 0.66131 70494 69622 33528 185186 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 constant187 C 10 {\displaystyle C_{10}} 0.12345 67891 01112 13141 188189Defined by concatenating representations of successive integers:

0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...

1933✗✗?
Salem constant190 σ 10 {\displaystyle \sigma _{10}} 1.17628 08182 59917 50654 191192Largest 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 193✗✓✓
Khinchin's constant194 K 0 {\displaystyle K_{0}} 2.68545 20010 65306 44530 195196 ∏ n = 1 ∞ [ 1 + 1 n ( n + 2 ) ] log 2 ⁡ ( n ) {\displaystyle \prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\log _{2}(n)}} 1934???
Lévy's constant (1)197 β {\displaystyle \beta } 1.18656 91104 15625 45282 198199 π 2 12 ln ⁡ 2 {\displaystyle {\frac {\pi ^{2}}{12\,\ln 2}}} 1935???
Lévy's constant (2)200 e β {\displaystyle e^{\beta }} 3.27582 29187 21811 15978 201202 e π 2 / ( 12 ln ⁡ 2 ) {\displaystyle e^{\pi ^{2}/(12\ln 2)}} 1936???
Copeland–Erdős constant203 C C E {\displaystyle {\mathcal {C}}_{CE}} 0.23571 11317 19232 93137 204205Defined by concatenating representations of successive prime numbers:

0.2 3 5 7 11 13 17 19 23 29 31 37 ...

1946 206✗??
Mills' constant207 A {\displaystyle A} 1.30637 78838 63080 69046 208209Smallest positive real number A such that ⌊ A 3 n ⌋ {\displaystyle \lfloor A^{3^{n}}\rfloor } is prime for all positive integers n1947???
Gompertz constant210 δ {\displaystyle \delta } 0.59634 73623 23194 07434 211212 ∫ 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 213???
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 214 π ln ⁡ 2 {\displaystyle {\frac {\pi }{\ln 2}}} Before 1958 215✗??
Magic angle216 θ m {\displaystyle \theta _{\mathrm {m} }} 0.95531 66181 245092 78163 217 arctan ⁡ 2 = arccos ⁡ 1 3 ≈ 54.7356 ∘ {\displaystyle \arctan {\sqrt {2}}=\arccos {\tfrac {1}{\sqrt {3}}}\approx \textstyle {54.7356}^{\circ }} Before 1959 218219✗✗✓
Artin's constant220 C A r t i n {\displaystyle C_{\mathrm {Artin} }} 0.37395 58136 19202 28805 221222 ∏ p  prime ( 1 − 1 p ( p − 1 ) ) {\displaystyle \prod _{p{\text{ prime}}}\left(1-{\frac {1}{p(p-1)}}\right)} Before 1961 223???
Porter's constant224 C {\displaystyle C} 1.46707 80794 33975 47289 225226 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 227???
Lochs constant228 L {\displaystyle L} 0.97027 01143 92033 92574 229230 6 ln ⁡ 2 ln ⁡ 10 π 2 {\displaystyle {\frac {6\ln 2\ln 10}{\pi ^{2}}}} 1964???
DeVicci's tesseract constant1.00743 47568 84279 37609 231The 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 232✗✓✓
Lieb's square ice constant2331.53960 07178 39002 03869 234235 ( 4 3 ) 3 2 = 8 3 3 {\displaystyle \left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}} 1967✗✓✓
Niven's constant236 C {\displaystyle C} 1.70521 11401 05367 76428 237238 1 + ∑ n = 2 ∞ ( 1 − 1 ζ ( n ) ) {\displaystyle 1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)}}\right)} 1969???
Stephens' constant2390.57595 99688 92945 43964 240241 ∏ p  prime ( 1 − p p 3 − 1 ) {\displaystyle \prod _{p{\text{ prime}}}\left(1-{\frac {p}{p^{3}-1}}\right)} 1969 242???
Regular paperfolding sequence243244 P {\displaystyle P} 0.85073 61882 01867 26036 245246 ∑ 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 247✗✗?
Reciprocal Fibonacci constant248 ψ {\displaystyle \psi } 3.35988 56662 43177 55317 249250 ∑ 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 nth Fibonacci number

1974 251✗??
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} [\lambda _{n,2}]}{n}}}

where E[λn,2] is the expected longest common subsequence of two random length-n binary strings

1975???
Feigenbaum constant δ252 δ {\displaystyle \delta } 4.66920 16091 02990 67185 253254 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 constants255 Ω {\displaystyle \Omega } In general they are uncomputable numbers.But one such number is 0.00787 49969 97812 3844.256257 ∑ 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 constant258 Δ ( 3 ) {\displaystyle \Delta (3)} 0.66170 71822 67176 23515 259260 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 261✗✗✓
Weierstrass constant2620.47494 93799 87920 65033 263264 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 1978265✗✗?
Fransén–Robinson constant266 F {\displaystyle F} 2.80777 02420 28519 36522 267268 ∫ 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 α269 α {\displaystyle \alpha } 2.50290 78750 95892 82228 270271Ratio between the width of a tine and the width of one of its two subtines in a bifurcation diagram1979???
Second du Bois-Reymond constant272 C 2 {\displaystyle C_{2}} 0.19452 80494 65325 11361 273274 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 275✗✗?
Erdős–Tenenbaum–Ford constant δ {\displaystyle \delta } 0.08607 13320 55934 20688 276 1 − 1 + log ⁡ log ⁡ 2 log ⁡ 2 {\displaystyle 1-{\frac {1+\log \log 2}{\log 2}}} 1984???
Conway's constant277 λ {\displaystyle \lambda } 1.30357 72690 34296 39125 278279Real 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 constant280 σ {\displaystyle \sigma } 0.35323 63718 54995 98454 281282 ∏ 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 283???
Backhouse's constant284 B {\displaystyle B} 1.45607 49485 82689 67139 285286 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 kth prime number

1995???
Viswanath constant2871.13198 82487 943 288289 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 1/21997???
Komornik–Loreti constant290 q {\displaystyle q} 1.78723 16501 82965 93301 291292Real 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 kth term of the Thue–Morse sequence

1998✗✗?
Embree–Trefethen constant β ⋆ {\displaystyle \beta ^{\star }} 0.702581999???
Heath-Brown–Moroz constant293 C {\displaystyle C} 0.00131 76411 54853 17810 294295 ∏ 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 296???
MRB constant297298299 S {\displaystyle S} 0.18785 96424 62067 12024 300301302 ∑ 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}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+\cdots } 1999???
Prime constant303 ρ {\displaystyle \rho } 0.41468 25098 51111 66024 304 ∑ 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 305✗??
Somos' quadratic recurrence constant306 σ {\displaystyle \sigma } 1.66168 79496 33594 12129 307308 ∏ 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 309???
Foias constant310 α {\displaystyle \alpha } 1.18745 23511 26501 05459 311312 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 disk3133140.59017 02995 08048 11302315316 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 317✗✗?
Taniguchi constant3180.67823 44919 17391 97803319320 ∏ 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 2005321???

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.

NameSymbolSetDecimal expansionContinued fractionNotes
Zero0 Z {\displaystyle \mathbb {Z} } 0.00000 00000[0; ]
Golomb–Dickman constant λ {\displaystyle \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, …]322E. Weisstein noted that the continued fraction has an unusually large number of 1s.323
Cahen's constant C 2 {\displaystyle C_{2}} R ∖ A {\displaystyle \mathbb {R} \setminus \mathbb {A} } 0.64341 05463[0; 1, 1, 1, 22, 32, 132, 1292, 252982, 4209841472, 2694251407415154862, …]324All 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 56649325[0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, …] 326327Using the continued fraction expansion, it was shown that if γ is rational, its denominator must exceed 10244663.
First continued fraction constant C 1 {\displaystyle C_{1}} R ∖ A {\displaystyle \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 ) {\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 55942328[0; 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, …] 329330Computed up to 4851389025 terms by E. Weisstein.331
One half⁠1/2⁠ Q {\displaystyle \mathbb {Q} } 0.50000 00000[0; 2]
Prouhet–Thue–Morse constant τ {\displaystyle \tau } R ∖ A {\displaystyle \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, …]332Infinitely many partial quotients are 4 or 5, and infinitely many partial quotients are greater than or equal to 50.333
Copeland–Erdős constant C C E {\displaystyle {\mathcal {C}}_{CE}} R ∖ Q {\displaystyle \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, …]334Computed 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.335
Base 10 Champernowne constant C 10 {\displaystyle C_{10}} R ∖ A {\displaystyle \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, …] 336Champernowne constants in any base exhibit sporadic large numbers; the 40th term in C 10 {\displaystyle C_{10}} has 2504 digits.
One1 N {\displaystyle \mathbb {N} } 1.00000 00000[1; ]
Phi, Golden ratio φ {\displaystyle \varphi } A {\displaystyle \mathbb {A} } 1.61803 39887337[1; 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …] 338The convergents are ratios of successive Fibonacci numbers.
Brun's constant B 2 {\displaystyle 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 nth roots of the denominators of the nth 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.339
Square root of 2 2 {\displaystyle {\sqrt {2}}} A {\displaystyle \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.
Two2 N {\displaystyle \mathbb {N} } 2.00000 00000[2; ]
Euler's number e {\displaystyle e} R ∖ A {\displaystyle \mathbb {R} \setminus \mathbb {A} } 2.71828 18285340[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, …] 341342The continued fraction expansion has the pattern [2; 1, 2, 1, 1, 4, 1, ..., 1, 2n, 1, ...].
Khinchin's constant K 0 {\displaystyle K_{0}} 2.68545 20011343[2; 1, 2, 5, 1, 1, 2, 1, 1, 3, 10, 2, 1, 3, 2, 24, 1, 3, 2, 3, 1, …] 344345For almost all real numbers x, the coefficients of the continued fraction of x have a finite geometric mean known as Khinchin's constant.
Three3 N {\displaystyle \mathbb {N} } 3.00000 00000[3; ]
Pi π {\displaystyle \pi } R ∖ A {\displaystyle \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, …] 346The 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

NameSymbolFormulaYearSet
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 constants347 γ n {\displaystyle \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 √γnn is the maximum of λ1(L) over all such lattices L.1822 to 1901 R {\displaystyle \mathbb {R} }
Hafner–Sarnak–McCurley constant348 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[1-\prod _{j=1}^{n}(1-p_{k}^{-j})\right]^{2}\right\}} 1883349 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 constants350351 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 Constant352 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 prime1919353 R {\displaystyle \mathbb {R} }
Champernowne constants354 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[\lambda _{n,k}]}{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

Site MathWorld Wolfram.com

Site OEIS.org

Site OEIS Wiki

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

References

  1. Weisstein, Eric W. "Constant". mathworld.wolfram.com. Retrieved 2020-08-08. https://mathworld.wolfram.com/Constant.html ↩

  2. Weisstein, Eric W. "Pi Formulas". MathWorld. /wiki/Eric_W._Weisstein ↩

  3. OEIS: A000796 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  4. Arndt & Haenel 2006, p. 167 - Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 2013-06-05. https://books.google.com/books?id=QwwcmweJCDQC ↩

  5. Hartl, Michael. "100,000 digits of Tau". Tau Day. Retrieved 22 January 2023. https://tauday.com/tau-digits ↩

  6. OEIS: A019692 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  7. Arndt & Haenel 2006, p. 167 - Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. ISBN 978-3-540-66572-4. Retrieved 2013-06-05. https://books.google.com/books?id=QwwcmweJCDQC ↩

  8. Calvin C Clawson (2001). Mathematical sorcery: revealing the secrets of numbers. Basic Books. p. IV. ISBN 978 0 7382 0496-3. 978 0 7382 0496-3 ↩

  9. Weisstein, Eric W. "Pythagoras's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  10. OEIS: A002193 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  11. Fowler and Robson, p. 368. Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection Archived 2012-08-13 at the Wayback Machine High resolution photographs, descriptions, and analysis of the root(2) tablet (YBC 7289) from the Yale Babylonian Collection http://it.stlawu.edu/%7Edmelvill/mesomath/tablets/YBC7289.html ↩

  12. Vijaya AV (2007). Figuring Out Mathematics. Dorling Kindcrsley (India) Pvt. Lid. p. 15. ISBN 978-81-317-0359-5. 978-81-317-0359-5 ↩

  13. Weisstein, Eric W. "Theodorus's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  14. OEIS: A002194 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  15. P A J Lewis (2008). Essential Mathematics 9. Ratna Sagar. p. 24. ISBN 9788183323673. 9788183323673 ↩

  16. OEIS: A002163 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  17. Timothy Gowers; June Barrow-Green; Imre Leade (2007). The Princeton Companion to Mathematics. Princeton University Press. p. 316. ISBN 978-0-691-11880-2. 978-0-691-11880-2 ↩

  18. Weisstein, Eric W. "Golden Ratio". MathWorld. /wiki/Eric_W._Weisstein ↩

  19. OEIS: A001622 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  20. Kapusta, Janos (2004), "The square, the circle, and the golden proportion: a new class of geometrical constructions" (PDF), Forma, 19: 293–313, archived from the original (PDF) on 2020-09-18, retrieved 2022-01-28. https://web.archive.org/web/20200918165840/http://www.scipress.org/journals/forma/pdf/1904/19040293.pdf ↩

  21. Weisstein, Eric W. "Silver Ratio". MathWorld. /wiki/Eric_W._Weisstein ↩

  22. OEIS: A014176 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  23. Kim Plofker (2009), Mathematics in India, Princeton University Press, ISBN 978-0-691-12067-6, pp. 54–56. /wiki/Mathematics_in_India_(book) ↩

  24. Weisstein, Eric W. "Delian Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  25. OEIS: A002580 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  26. Plutarch. "718ef". Quaestiones convivales VIII.ii. Archived from the original on 2009-11-19. Retrieved 2019-05-24. And therefore Plato himself dislikes Eudoxus, Archytas, and Menaechmus for endeavoring to bring down the doubling the cube to mechanical operations https://web.archive.org/web/20091119061142/http://ebooks.adelaide.edu.au/p/plutarch/symposiacs/chapter8.html#section80 ↩

  27. OEIS: A002581 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  28. Christensen, Thomas (2002), The Cambridge History of Western Music Theory, Cambridge University Press, p. 205, ISBN 978-0521686983 978-0521686983 ↩

  29. OEIS: A010774 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  30. Koshy, Thomas (2017). Fibonacci and Lucas Numbers with Applications (2 ed.). John Wiley & Sons. ISBN 9781118742174. Retrieved 14 August 2018. 9781118742174 ↩

  31. OEIS: A092526 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  32. Keith J. Devlin (1999). Mathematics: The New Golden Age. Columbia University Press. p. 66. ISBN 978-0-231-11638-1. 978-0-231-11638-1 ↩

  33. 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. /wiki/Imaginary_unit ↩

  34. Mireille Bousquet-Mélou. Two-dimensional self-avoiding walks (PDF). CNRS, LaBRI, Bordeaux, France. /wiki/Mireille_Bousquet-M%C3%A9lou ↩

  35. Hugo Duminil-Copin & Stanislav Smirnov (2011). The connective constant of the honeycomb lattice √ (2 + √ 2) (PDF). Université de Geneve. http://www.unige.ch/~smirnov/slides/slides-saw.pdf ↩

  36. Weisstein, Eric W. "Self-Avoiding Walk Connective Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  37. OEIS: A179260 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  38. OEIS: A179260 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  39. Richard J. Mathar (2013). "Circumscribed Regular Polygons". arXiv:1301.6293 [math.MG]. /wiki/ArXiv_(identifier) ↩

  40. Weisstein, Eric W. "Polygon Inscribing". MathWorld. /wiki/Eric_W._Weisstein ↩

  41. OEIS: A085365 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  42. OEIS: A085365 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  43. Weisstein, Eric W. "Wallis's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  44. OEIS: A007493 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  45. E.Kasner y J.Newman. (2007). Mathematics and the Imagination. Conaculta. p. 77. ISBN 978-968-5374-20-0. 978-968-5374-20-0 ↩

  46. Weisstein, Eric W. "e". MathWorld. /wiki/Eric_W._Weisstein ↩

  47. OEIS: A001113 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  48. O'Connor, J J; Robertson, E F. "The number e". MacTutor History of Mathematics. http://www-history.mcs.st-and.ac.uk/HistTopics/e.html ↩

  49. Annie Cuyt; Vigdis Brevik Petersen; Brigitte Verdonk; Haakon Waadeland; William B. Jones (2008). Handbook of Continued Fractions for Special Functions. Springer. p. 182. ISBN 978-1-4020-6948-2. 978-1-4020-6948-2 ↩

  50. Weisstein, Eric W. "Natural Logarithm of 2". MathWorld. /wiki/Eric_W._Weisstein ↩

  51. OEIS: A002162 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  52. Cajori, Florian (1991). A History of Mathematics (5th ed.). AMS Bookstore. p. 152. ISBN 0-8218-2102-4. 0-8218-2102-4 ↩

  53. O'Connor, J. J.; Robertson, E. F. (September 2001). "The number e". The MacTutor History of Mathematics archive. Retrieved 2009-02-02. http://www-history.mcs.st-and.ac.uk/HistTopics/e.html ↩

  54. J. Coates; Martin J. Taylor (1991). L-Functions and Arithmetic. Cambridge University Press. p. 333. ISBN 978-0-521-38619-7. 978-0-521-38619-7 ↩

  55. Weisstein, Eric W. "Lemniscate Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  56. OEIS: A062539 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  57. Weisstein, Eric W. "Euler–Mascheroni Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  58. OEIS: A001620 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  59. Robert Baillie (2013). "Summing The Curious Series of Kempner and Irwin". arXiv:0806.4410 [math.CA]. /wiki/ArXiv_(identifier) ↩

  60. Weisstein, Eric W. "Erdos-Borwein Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  61. OEIS: A065442 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  62. Leonhard Euler (1749). Consideratio quarumdam serierum, quae singularibus proprietatibus sunt praeditae. p. 108. http://www.math.dartmouth.edu/~euler/pages/E190.html ↩

  63. Weisstein, Eric W. "Omega Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  64. OEIS: A030178 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  65. Annie Cuyt; Vigdis Brevik Petersen; Brigitte Verdonk; Haakon Waadelantl; William B. Jones. (2008). Handbook of Continued Fractions for Special Functions. Springer. p. 188. ISBN 978-1-4020-6948-2. 978-1-4020-6948-2 ↩

  66. Weisstein, Eric W. "Apéry's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  67. OEIS: A002117 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  68. OEIS: A002117 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  69. Howard Curtis (2014). Orbital Mechanics for Engineering Students. Elsevier. p. 159. ISBN 978-0-08-097747-8. 978-0-08-097747-8 ↩

  70. Weisstein, Eric W. "Laplace Limit". MathWorld. /wiki/Eric_W._Weisstein ↩

  71. OEIS: A033259 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  72. Johann Georg Soldner (1809). Théorie et tables d'une nouvelle fonction transcendante (in French). J. Lindauer, München. p. 42. https://archive.org/details/bub_gb_g4Q_AAAAcAAJ ↩

  73. Lorenzo Mascheroni (1792). Adnotationes ad calculum integralem Euleri (in Latin). Petrus Galeatius, Ticini. p. 17. https://archive.org/details/bub_gb_XkgDAAAAQAAJ ↩

  74. Weisstein, Eric W. "Soldner's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  75. OEIS: A070769 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  76. OEIS: A070769 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  77. Keith B. Oldham; Jan C. Myland; Jerome Spanier (2009). An Atlas of Functions: With Equator, the Atlas Function Calculator. Springer. p. 15. ISBN 978-0-387-48806-6. 978-0-387-48806-6 ↩

  78. Weisstein, Eric W. "Gauss's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  79. OEIS: A014549 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  80. Nielsen, Mikkel Slot. (July 2016). Undergraduate convexity : problems and solutions. World Scientific. p. 162. ISBN 9789813146211. OCLC 951172848. 9789813146211 ↩

  81. Steven Finch (2014). Errata and Addenda to Mathematical Constants (PDF). Harvard.edu. Archived from the original (PDF) on 2016-03-16. Retrieved 2013-12-17. https://web.archive.org/web/20160316175639/http://www.people.fas.harvard.edu/~sfinch/csolve/erradd.pdf ↩

  82. Weisstein, Eric W. "Hermite Constants". MathWorld. /wiki/Eric_W._Weisstein ↩

  83. OEIS: A246724 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  84. Calvin C. Clawson (2003). Mathematical Traveler: Exploring the Grand History of Numbers. Perseus. p. 187. ISBN 978-0-7382-0835-0. 978-0-7382-0835-0 ↩

  85. Weisstein, Eric W. "Liouville's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  86. OEIS: A012245 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  87. Weisstein, Eric W. "Continued Fraction Constants". MathWorld. /wiki/Eric_W._Weisstein ↩

  88. OEIS: A052119 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  89. Waldschmidt, Michel (2021). "Irrationality and transcendence of values of special functions" (PDF). https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/ValuesSpecialFunctions.pdf ↩

  90. Amoretti, F. (1855). "Sur la fraction continue [0,1,2,3,4,...]". Nouvelles annales de mathématiques. 1 (14): 40–44. http://www.numdam.org/item/?id=NAM_1855_1_14__40_1 ↩

  91. L. J. Lloyd James Peter Kilford (2008). Modular Forms: A Classical and Computational Introduction. Imperial College Press. p. 107. ISBN 978-1-84816-213-6. 978-1-84816-213-6 ↩

  92. Weisstein, Eric W. "Ramanujan Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  93. OEIS: A060295 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  94. Weisstein, Eric W. "Glaisher-Kinkelin Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  95. OEIS: A074962 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  96. OEIS: A074962 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  97. Henri Cohen (2000). Number Theory: Volume II: Analytic and Modern Tools. Springer. p. 127. ISBN 978-0-387-49893-5. 978-0-387-49893-5 ↩

  98. H. M. Srivastava; Choi Junesang (2001). Series Associated With the Zeta and Related Functions. Kluwer Academic Publishers. p. 30. ISBN 978-0-7923-7054-3. 978-0-7923-7054-3 ↩

  99. E. Catalan (1864). Mémoire sur la transformation des séries, et sur quelques intégrales définies, Comptes rendus hebdomadaires des séances de l'Académie des sciences 59. Kluwer Academic éditeurs. p. 618. https://books.google.com/books?id=LXZFAAAAcAAJ&pg=PA618 ↩

  100. Weisstein, Eric W. "Catalan's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  101. OEIS: A006752 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  102. James Stewart (2010). Single Variable Calculus: Concepts and Contexts. Brooks/Cole. p. 314. ISBN 978-0-495-55972-6. 978-0-495-55972-6 ↩

  103. Weisstein, Eric W. "Dottie Number". MathWorld. /wiki/Eric_W._Weisstein ↩

  104. OEIS: A003957 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  105. Weisstein, Eric W. "Dottie Number". MathWorld. /wiki/Eric_W._Weisstein ↩

  106. Julian Havil (2003). Gamma: Exploring Euler's Constant. Princeton University Press. p. 64. ISBN 9780691141336. 9780691141336 ↩

  107. Weisstein, Eric W. "Mertens Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  108. OEIS: A077761 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  109. Steven Finch (2014). Errata and Addenda to Mathematical Constants (PDF). Harvard.edu. p. 59. Archived from the original (PDF) on 2016-03-16. Retrieved 2013-12-17. https://web.archive.org/web/20160316175639/http://www.people.fas.harvard.edu/~sfinch/csolve/erradd.pdf ↩

  110. Weisstein, Eric W. "Universal Parabolic Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  111. OEIS: A103710 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  112. Osborne, George Abbott (1891). An Elementary Treatise on the Differential and Integral Calculus. Leach, Shewell, and Sanborn. pp. 250. https://archive.org/details/anelementarytre00osbogoog ↩

  113. Yann Bugeaud (2004). Series representations for some mathematical constants. Cambridge University Press. p. 72. ISBN 978-0-521-82329-6. 978-0-521-82329-6 ↩

  114. Weisstein, Eric W. "Cahen's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  115. OEIS: A118227 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  116. David Wells (1997). The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books Ltd. p. 4. ISBN 9780141929408. 9780141929408 ↩

  117. Weisstein, Eric W. "Gelfonds Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  118. OEIS: A039661 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  119. Tijdeman, Robert (1976). "On the Gel'fond–Baker method and its applications". In Felix E. Browder (ed.). Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. Vol. XXVIII.1. American Mathematical Society. pp. 241–268. ISBN 0-8218-1428-1. Zbl 0341.10026. 0-8218-1428-1 ↩

  120. David Cohen (2006). Precalculus: With Unit Circle Trigonometry. Thomson Learning Inc. p. 328. ISBN 978-0-534-40230-3. 978-0-534-40230-3 ↩

  121. Weisstein, Eric W. "Gelfond-Schneider Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  122. OEIS: A007507 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  123. OEIS: A007507 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  124. Helmut Brass; Knut Petras (2010). Quadrature Theory: The Theory of Numerical Integration on a Compact Interval. AMS. p. 274. ISBN 978-0-8218-5361-0. 978-0-8218-5361-0 ↩

  125. Weisstein, Eric W. "Favard Constants". MathWorld. /wiki/Eric_W._Weisstein ↩

  126. OEIS: A111003 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  127. Ángulo áureo. http://fibonacci.ucoz.com/index/ang/0-9 ↩

  128. Weisstein, Eric W. "Golden Angle". MathWorld. /wiki/Eric_W._Weisstein ↩

  129. OEIS: A131988 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  130. Eric W. Weisstein (2002). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 1356. ISBN 9781420035223. 9781420035223 ↩

  131. Weisstein, Eric W. "Sierpinski Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  132. OEIS: A062089 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  133. Richard E. Crandall; Carl B. Pomerance (2005). Prime Numbers: A Computational Perspective. Springer. p. 80. ISBN 978-0387-25282-7. 978-0387-25282-7 ↩

  134. Weisstein, Eric W. "Landau-Ramanujan Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  135. OEIS: A064533 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  136. OEIS: A064533 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  137. Mauro Fiorentini. Nielsen – Ramanujan (costanti di). http://bitman.name/math/article/872 ↩

  138. Weisstein, Eric W. "Nielsen-Ramanujan Constants". MathWorld. /wiki/Eric_W._Weisstein ↩

  139. OEIS: A072691 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  140. Steven Finch. Volumes of Hyperbolic 3-Manifolds (PDF). Harvard University. Archived from the original (PDF) on 2015-09-19. https://web.archive.org/web/20150919160427/http://www.people.fas.harvard.edu/~sfinch/csolve/hyp.pdf ↩

  141. Weisstein, Eric W. "Gieseking's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  142. OEIS: A143298 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  143. Lloyd N. Trefethen (2013). Approximation Theory and Approximation Practice. SIAM. p. 211. ISBN 978-1-611972-39-9. 978-1-611972-39-9 ↩

  144. Weisstein, Eric W. "Bernstein's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  145. OEIS: A073001 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  146. Agronomof, M. (1914). "Sur une suite récurrente". Mathesis. 4: 125–126. ↩

  147. Weisstein, Eric W. "Tribonacci Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  148. OEIS: A058265 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  149. Thomas Koshy (2007). Elementary Number Theory with Applications. Elsevier. p. 119. ISBN 978-0-12-372-487-8. 978-0-12-372-487-8 ↩

  150. Weisstein, Eric W. "Brun's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  151. OEIS: A065421 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  152. OEIS: A065421 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  153. Weisstein, Eric W. "Twin Primes Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  154. OEIS: A005597 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  155. Ian Stewart (1996). Professor Stewart's Cabinet of Mathematical Curiosities. Birkhäuser Verlag. ISBN 978-1-84765-128-0. 978-1-84765-128-0 ↩

  156. Weisstein, Eric W. "Plastic Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  157. OEIS: A060006 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  158. OEIS: A060006 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  159. Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 1688. ISBN 978-1-58488-347-0. 978-1-58488-347-0 ↩

  160. Weisstein, Eric W. "Bloch Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  161. OEIS: A085508 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  162. OEIS: A085508 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  163. Rees, DG (1987), Foundations of Statistics, CRC Press, p. 246, ISBN 0-412-28560-6, Why 95% confidence? Why not some other confidence level? The use of 95% is partly convention, but levels such as 90%, 98% and sometimes 99.9% are also used. 0-412-28560-6 ↩

  164. "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. https://web.archive.org/web/20080205120031/http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm ↩

  165. 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. 0-8251-3863-9 ↩

  166. Swift, MB (2009). "Comparison of Confidence Intervals for a Poisson Mean – Further Considerations". Communications in Statistics – Theory and Methods. 38 (5): 748–759. doi:10.1080/03610920802255856. S2CID 120748700. In modern applied practice, almost all confidence intervals are stated at the 95% level. /wiki/Doi_(identifier) ↩

  167. Weisstein, Eric W. "Confidence Interval". MathWorld. /wiki/Eric_W._Weisstein ↩

  168. OEIS: A220510 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  169. Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 1688. ISBN 978-1-58488-347-0. 978-1-58488-347-0 ↩

  170. Weisstein, Eric W. "Landau Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  171. OEIS: A081760 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  172. Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 1688. ISBN 978-1-58488-347-0. 978-1-58488-347-0 ↩

  173. Steven Finch (2014). Errata and Addenda to Mathematical Constants (PDF). Harvard.edu. p. 53. Archived from the original (PDF) on 2016-03-16. Retrieved 2013-12-17. https://web.archive.org/web/20160316175639/http://www.people.fas.harvard.edu/~sfinch/csolve/erradd.pdf ↩

  174. Weisstein, Eric W. "Thue-Morse Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  175. OEIS: A014571 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  176. OEIS: A014571 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  177. Eric W. Weisstein (2002). CRC Concise Encyclopedia of Mathematics. Crc Press. p. 1212. ISBN 9781420035223. 9781420035223 ↩

  178. Weisstein, Eric W. "Golomb–Dickman Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  179. OEIS: A084945 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  180. Horst Alzer (2002). "Journal of Computational and Applied Mathematics, Volume 139, Issue 2" (PDF). Journal of Computational and Applied Mathematics. 139 (2): 215–230. doi:10.1016/S0377-0427(01)00426-5. http://ac.els-cdn.com/S0377042701004265/1-s2.0-S0377042701004265-main.pdf?_tid=c20cf466-f4bf-11e3-bd9c-00000aacb362&acdnat=1402859198_57de7868bcc50086f092c66898ec6a33 ↩

  181. Weisstein, Eric W. "Lebesgue Constants". MathWorld. /wiki/Eric_W._Weisstein ↩

  182. OEIS: A243277 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  183. Weisstein, Eric W. "Lebesgue Constants". MathWorld. /wiki/Eric_W._Weisstein ↩

  184. ECKFORD COHEN (1962). SOME ASYMPTOTIC FORMULAS IN THE THEORY OF NUMBERS (PDF). University of Tennessee. p. 220. https://www.ams.org/journals/tran/1964-112-02/S0002-9947-1964-0166181-5/S0002-9947-1964-0166181-5.pdf ↩

  185. Weisstein, Eric W. "Feller–Tornier Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  186. OEIS: A065493 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  187. Michael J. Dinneen; Bakhadyr Khoussainov; Prof. Andre Nies (2012). Computation, Physics and Beyond. Springer. p. 110. ISBN 978-3-642-27653-8. 978-3-642-27653-8 ↩

  188. Weisstein, Eric W. "Champernowne Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  189. OEIS: A033307 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  190. Pei-Chu Hu, Chung-Chun (2008). Distribution Theory of Algebraic Numbers. Hong Kong University. p. 246. ISBN 978-3-11-020536-7. 978-3-11-020536-7 ↩

  191. Weisstein, Eric W. "Salem Constants". MathWorld. /wiki/Eric_W._Weisstein ↩

  192. OEIS: A073011 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  193. OEIS: A073011 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  194. Julian Havil (2003). Gamma: Exploring Euler's Constant. Princeton University Press. p. 161. ISBN 9780691141336. 9780691141336 ↩

  195. Weisstein, Eric W. "Khinchin's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  196. OEIS: A002210 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  197. Aleksandr I͡Akovlevich Khinchin (1997). Continued Fractions. Courier Dover Publications. p. 66. ISBN 978-0-486-69630-0. 978-0-486-69630-0 ↩

  198. Weisstein, Eric W. "Levy Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  199. OEIS: A100199 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  200. Marek Wolf (2018). "Two arguments that the nontrivial zeros of the Riemann zeta function are irrational". Computational Methods in Science and Technology. 24 (4): 215–220. arXiv:1002.4171. doi:10.12921/cmst.2018.0000049. S2CID 115174293. /wiki/ArXiv_(identifier) ↩

  201. Weisstein, Eric W. "Levy Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  202. OEIS: A086702 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  203. Yann Bugeaud (2012). Distribution Modulo One and Diophantine Approximation. Cambridge University Press. p. 87. ISBN 978-0-521-11169-0. 978-0-521-11169-0 ↩

  204. Weisstein, Eric W. "Copeland–Erdos Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  205. OEIS: A033308 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  206. OEIS: A033308 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  207. Laith Saadi (2004). Stealth Ciphers. Trafford Publishing. p. 160. ISBN 978-1-4120-2409-9. 978-1-4120-2409-9 ↩

  208. Weisstein, Eric W. "Mills Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  209. OEIS: A051021 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  210. Annie Cuyt; Viadis Brevik Petersen; Brigitte Verdonk; William B. Jones (2008). Handbook of continued fractions for special functions. Springer Science. p. 190. ISBN 978-1-4020-6948-2. 978-1-4020-6948-2 ↩

  211. Weisstein, Eric W. "Gompertz Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  212. OEIS: A073003 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  213. OEIS: A073003 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  214. OEIS: A163973 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  215. OEIS: A163973 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  216. Andras Bezdek (2003). Discrete Geometry. Marcel Dekkcr, Inc. p. 150. ISBN 978-0-8247-0968-6. 978-0-8247-0968-6 ↩

  217. OEIS: A195696 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  218. Lowe, I. J. (1959-04-01). "Free Induction Decays of Rotating Solids". Physical Review Letters. 2 (7): 285–287. Bibcode:1959PhRvL...2..285L. doi:10.1103/PhysRevLett.2.285. ISSN 0031-9007. https://link.aps.org/doi/10.1103/PhysRevLett.2.285 ↩

  219. Andras Bezdek (2003). Discrete Geometry. Marcel Dekkcr, Inc. p. 150. ISBN 978-0-8247-0968-6. 978-0-8247-0968-6 ↩

  220. Paulo Ribenboim (2000). My Numbers, My Friends: Popular Lectures on Number Theory. Springer. p. 66. ISBN 978-0-387-98911-2. 978-0-387-98911-2 ↩

  221. Weisstein, Eric W. "Artin's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  222. OEIS: A005596 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  223. OEIS: A005596 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  224. Michel A. Théra (2002). Constructive, Experimental, and Nonlinear Analysis. CMS-AMS. p. 77. ISBN 978-0-8218-2167-1. 978-0-8218-2167-1 ↩

  225. Weisstein, Eric W. "Porter's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  226. OEIS: A086237 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  227. OEIS: A086237 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  228. Steven Finch (2007). Continued Fraction Transformation (PDF). Harvard University. p. 7. Archived from the original (PDF) on 2016-04-19. Retrieved 2015-02-28. https://web.archive.org/web/20160419114446/http://www.people.fas.harvard.edu/~sfinch/csolve/kz.pdf ↩

  229. Weisstein, Eric W. "Lochs' Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  230. OEIS: A086819 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  231. OEIS: A243309 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  232. OEIS: A243309 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  233. Robin Whitty. Lieb's Square Ice Theorem (PDF). http://www.theoremoftheday.org/MathPhysics/Lieb/TotDLieb.pdf ↩

  234. Weisstein, Eric W. "Liebs Square Ice Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  235. OEIS: A118273 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  236. Ivan Niven. Averages of exponents in factoring integers (PDF). https://www.ams.org/journals/proc/1969-022-02/S0002-9939-1969-0241373-5/S0002-9939-1969-0241373-5.pdf ↩

  237. Weisstein, Eric W. "Niven's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  238. OEIS: A033150 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  239. Steven Finch (2005). Class Number Theory (PDF). Harvard University. p. 8. Archived from the original (PDF) on 2016-04-19. Retrieved 2014-04-15. https://web.archive.org/web/20160419150530/http://www.people.fas.harvard.edu/~sfinch/csolve/clss.pdf ↩

  240. Weisstein, Eric W. "Stephen's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  241. OEIS: A065478 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  242. OEIS: A065478 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  243. Francisco J. Aragón Artacho; David H. Baileyy; Jonathan M. Borweinz; Peter B. Borwein (2012). Tools for visualizing real numbers (PDF). p. 33. Archived from the original (PDF) on 2017-02-20. Retrieved 2014-01-20. https://web.archive.org/web/20170220175429/https://carma.newcastle.edu.au/jon/tools1.pdf ↩

  244. Papierfalten (PDF). 1998. http://www.jgiesen.de/Divers/PapierFalten/PapierFalten.pdf ↩

  245. Weisstein, Eric W. "Paper Folding Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  246. OEIS: A143347 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  247. OEIS: A143347 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  248. Gérard P. Michon (2005). Numerical Constants. Numericana. http://www.numericana.com/answer/constants.htm#prevost ↩

  249. Weisstein, Eric W. "Reciprocal Fibonacci Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  250. OEIS: A079586 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  251. OEIS: A079586 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  252. Kathleen T. Alligood (1996). Chaos: An Introduction to Dynamical Systems. Springer. ISBN 978-0-387-94677-1. 978-0-387-94677-1 ↩

  253. Weisstein, Eric W. "Feigenbaum Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  254. OEIS: A006890 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  255. David Darling (2004). The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes. Wiley & Sons inc. p. 63. ISBN 978-0-471-27047-8. 978-0-471-27047-8 ↩

  256. Weisstein, Eric W. "Chaitin's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  257. OEIS: A100264 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  258. Steven R. Finch (2003). Mathematical Constants. Cambridge University Press. p. 479. ISBN 978-3-540-67695-9. Schmutz. 978-3-540-67695-9 ↩

  259. Weisstein, Eric W. "Robbins Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  260. OEIS: A073012 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  261. OEIS: A073012 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  262. Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 151. ISBN 978-1-58488-347-0. 978-1-58488-347-0 ↩

  263. Weisstein, Eric W. "Weierstrass Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  264. OEIS: A094692 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  265. Waldschmidt, M. "Nombres transcendants et fonctions sigma de Weierstrass." C. R. Math. Rep. Acad. Sci. Canada 1, 111-114, 1978/79. ↩

  266. Dusko Letic; Nenad Cakic; Branko Davidovic; Ivana Berkovic. Orthogonal and diagonal dimension fluxes of hyperspherical function (PDF). Springer. http://www.advancesindifferenceequations.com/content/pdf/1687-1847-2012-22.pdf ↩

  267. Weisstein, Eric W. "Fransen-Robinson Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  268. OEIS: A058655 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  269. K. T. Chau; Zheng Wang (201). Chaos in Electric Drive Systems: Analysis, Control and Application. John Wiley & Son. p. 7. ISBN 978-0-470-82633-1. {{cite book}}: ISBN / Date incompatibility (help) 978-0-470-82633-1 ↩

  270. Weisstein, Eric W. "Feigenbaum Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  271. OEIS: A006891 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  272. Steven R. Finch (2003). Mathematical Constants. Cambridge University Press. p. 238. ISBN 978-3-540-67695-9. 978-3-540-67695-9 ↩

  273. Weisstein, Eric W. "du Bois-Reymond Constants". MathWorld. /wiki/Eric_W._Weisstein ↩

  274. OEIS: A062546 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  275. OEIS: A062546 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  276. OEIS: A074738 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  277. Facts On File, Incorporated (1997). Mathematics Frontiers. Infobase. p. 46. ISBN 978-0-8160-5427-5. 978-0-8160-5427-5 ↩

  278. Weisstein, Eric W. "Conway's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  279. OEIS: A014715 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  280. Steven R. Finch (2003). Mathematical Constants. Cambridge University Press. p. 110. ISBN 978-3-540-67695-9. 978-3-540-67695-9 ↩

  281. Weisstein, Eric W. "Hafner-Sarnak-McCurley Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  282. OEIS: A085849 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  283. OEIS: A085849 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  284. Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 151. ISBN 978-1-58488-347-0. 978-1-58488-347-0 ↩

  285. Weisstein, Eric W. "Backhouse's Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  286. OEIS: A072508 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  287. DIVAKAR VISWANATH (1999). RANDOM FIBONACCI SEQUENCES AND THE NUMBER 1.13198824... (PDF). MATHEMATICS OF COMPUTATION. https://www.ams.org/journals/mcom/2000-69-231/S0025-5718-99-01145-X/S0025-5718-99-01145-X.pdf ↩

  288. Weisstein, Eric W. "Random Fibonacci Sequence". MathWorld. /wiki/Eric_W._Weisstein ↩

  289. OEIS: A078416 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  290. Christoph Lanz. k-Automatic Reals (PDF). Technischen Universität Wien. http://dmg.tuwien.ac.at/drmota/DiplomarbeitLanz.pdf ↩

  291. Weisstein, Eric W. "Komornik-Loreti Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  292. OEIS: A055060 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  293. J. B. Friedlander; A. Perelli; C. Viola; D.R. Heath-Brown; H.Iwaniec; J. Kaczorowski (2002). Analytic Number Theory. Springer. p. 29. ISBN 978-3-540-36363-7. 978-3-540-36363-7 ↩

  294. Weisstein, Eric W. "Heath-Brown-Moroz Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  295. OEIS: A118228 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  296. OEIS: A118228 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  297. Richard E. Crandall (2012). Unified algorithms for polylogarithm, L-series, and zeta variants (PDF). perfscipress.com. Archived from the original on 2013-04-30. https://web.archive.org/web/20130430193005/http://www.perfscipress.com/papers/UniversalTOC25.pdf ↩

  298. RICHARD J. MATHAR (2010). "NUMERICAL EVALUATION OF THE OSCILLATORY INTEGRAL OVER exp(I pi x)x^1/x BETWEEN 1 AND INFINITY". arXiv:0912.3844 [math.CA]. /wiki/ArXiv_(identifier) ↩

  299. M.R.Burns (1999). Root constant. Marvin Ray Burns. http://marvinrayburns.com/Original_MRB_Post.html ↩

  300. Weisstein, Eric W. "MRB Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  301. MRB constant //oeis.org/wiki/MRB_constant ↩

  302. OEIS: A037077 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  303. Hardy, G. H. (2008). An introduction to the theory of numbers. E. M. Wright, D. R. Heath-Brown, Joseph H. Silverman (6th ed.). Oxford: Oxford University Press. ISBN 978-0-19-921985-8. OCLC 214305907. 978-0-19-921985-8 ↩

  304. OEIS: A051006 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  305. OEIS: A051006 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  306. Jesus Guillera; Jonathan Sondow (2008). "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent". The Ramanujan Journal. 16 (3): 247–270. arXiv:math/0506319. doi:10.1007/s11139-007-9102-0. S2CID 119131640. /wiki/ArXiv_(identifier) ↩

  307. Weisstein, Eric W. "Somos's Quadratic Recurrence Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  308. OEIS: A112302 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  309. Weisstein, Eric W. "Somos's Quadratic Recurrence Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  310. Andrei Vernescu (2007). Gazeta Matemetica Seria a revista de cultur Matemetica Anul XXV(CIV)Nr. 1, Constante de tip Euler generalízate (PDF). p. 14. http://ssmr.ro/gazeta/gma/2007/gma-1-2007.pdf ↩

  311. Weisstein, Eric W. "Foias Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  312. OEIS: A085848 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  313. Steven Finch (2014). Electrical Capacitance (PDF). Harvard.edu. p. 1. Archived from the original (PDF) on 2016-04-19. Retrieved 2015-10-12. https://web.archive.org/web/20160419150944/http://www.people.fas.harvard.edu/~sfinch/csolve/capa.pdf ↩

  314. Ransford, Thomas (2010). "Computation of logarithmic capacity". Computational Methods and Function Theory. 10 (2): 555–578. doi:10.1007/BF03321780. MR 2791324. /wiki/Doi_(identifier) ↩

  315. Weisstein, Eric W. "Logarithmic Capacity". MathWorld. /wiki/Eric_W._Weisstein ↩

  316. OEIS: A249205 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  317. OEIS: A249205 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  318. Steven Finch (2005). Class Number Theory (PDF). Harvard University. p. 8. Archived from the original (PDF) on 2016-04-19. Retrieved 2014-04-15. https://web.archive.org/web/20160419150530/http://www.people.fas.harvard.edu/~sfinch/csolve/clss.pdf ↩

  319. Weisstein, Eric W. "Taniguchis Constant". MathWorld. /wiki/Eric_W._Weisstein ↩

  320. OEIS: A175639 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  321. Steven Finch (2005). Class Number Theory (PDF). Harvard University. p. 8. Archived from the original (PDF) on 2016-04-19. Retrieved 2014-04-15. https://web.archive.org/web/20160419150530/http://www.people.fas.harvard.edu/~sfinch/csolve/clss.pdf ↩

  322. OEIS: A225336 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  323. Weisstein, Eric W. "Golomb-Dickman Constant Continued Fraction". MathWorld. /wiki/Eric_W._Weisstein ↩

  324. OEIS: A006280 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  325. Cuyt et al. 2008, p. 182. - 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. ↩

  326. Cuyt et al. 2008, p. 182. - 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. ↩

  327. OEIS: A002852 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  328. Borwein et al. 2014, p. 190. - 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. https://search.worldcat.org/issn/0950-2815 ↩

  329. Borwein et al. 2014, p. 190. - 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. https://search.worldcat.org/issn/0950-2815 ↩

  330. OEIS: A014538 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  331. Weisstein, Eric W. "Catalan's Constant Continued Fraction". MathWorld. /wiki/Eric_W._Weisstein ↩

  332. OEIS: A014572 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  333. Bugeaud, Yann; Queffélec, Martine (2013). "On Rational Approximation of the Binary Thue-Morse-Mahler Number". Journal of Integer Sequences. 16 (13.2.3). https://cs.uwaterloo.ca/journals/JIS/VOL16/Bugeaud/bugeaud3.html ↩

  334. OEIS: A030168 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  335. Weisstein, Eric W. "Copeland–Erdős Constant Continued Fraction". MathWorld. /wiki/Eric_W._Weisstein ↩

  336. OEIS: A030167 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  337. Cuyt et al. 2008, p. 185. - 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. ↩

  338. Cuyt et al. 2008, p. 186. - 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. ↩

  339. Wolf, Marek (22 February 2010). "Remark on the irrationality of the Brun's constant". arXiv:1002.4174 [math.NT]. /wiki/ArXiv_(identifier) ↩

  340. Cuyt et al. 2008, p. 176. - 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. ↩

  341. Cuyt et al. 2008, p. 179. - 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. ↩

  342. OEIS: A003417 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  343. Cuyt et al. 2008, p. 190. - 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. ↩

  344. Cuyt et al. 2008, p. 191. - 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. ↩

  345. OEIS: A002211 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  346. OEIS: A001203 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  347. "Hermite Constants". https://mathworld.wolfram.com/HermiteConstants.html ↩

  348. Holger Hermanns; Roberto Segala (2000). Process Algebra and Probabilistic Methods. Springer-Verlag. p. 270. ISBN 978-3-540-67695-9. 978-3-540-67695-9 ↩

  349. Weisstein, Eric W. "Relatively Prime". MathWorld. /wiki/Eric_W._Weisstein ↩

  350. Helmut Brass; Knut Petras (2010). Quadrature Theory: The Theory of Numerical Integration on a Compact Interval. AMS. p. 274. ISBN 978-0-8218-5361-0. 978-0-8218-5361-0 ↩

  351. "Favard Constants". https://mathworld.wolfram.com/FavardConstants.html ↩

  352. Thomas Koshy (2007). Elementary Number Theory with Applications. Elsevier. p. 119. ISBN 978-0-12-372-487-8. 978-0-12-372-487-8 ↩

  353. OEIS: A065421 /wiki/On-Line_Encyclopedia_of_Integer_Sequences ↩

  354. Michael J. Dinneen; Bakhadyr Khoussainov; Prof. Andre Nies (2012). Computation, Physics and Beyond. Springer. p. 110. ISBN 978-3-642-27653-8. 978-3-642-27653-8 ↩