List of special numbers

This list of special numbers lists numbers that have one or more conspicuous mathematical properties and numbers that have a special cultural or technical meaning. The latter numbers are listed in the second part of this article.

Numbers with special mathematical properties

Until 0

• −2
  • Smallest whole number for which the ring is Euclidean . ${\displaystyle d}$ ${\displaystyle \mathbb {Z} [{\sqrt {d}}]}$
  • Largest trivial zero of the zeta function . ${\displaystyle \zeta (-2)=0}$
• −1
  • A unit in the ring of whole numbers and its extension rings.
  • Single complex number of the multiplicative order . ${\displaystyle 2}$
  • In the body of complex numbers is ${\displaystyle -1={\mathrm {i} }^{2}=e^{\pi {\mathrm {i} }}}$
  • smallest number occurring as a dimension (namely sometimes the empty set)
• −0.5
  • Functional value of the zeta function ${\displaystyle \zeta (0)}$
• −0.083333333333333 ...
  • Functional value of the zeta function ${\displaystyle \zeta (-1)=-{\tfrac {1}{12}}}$
• 0
  • Neutral element of addition in the ring of whole numbers and its expansion rings. (These include the fields of rational , real and complex numbers .)
  • "Zero element" of the multiplication (ie, if there is a factor , so is the product) ${\displaystyle 0}$
  • only number for which the function has a point of discontinuity (if the definition is followed) ${\displaystyle n}$ ${\displaystyle n^{x}}$ ${\displaystyle 0^{0}=1}$
  • first index of some countably indexed series, but usually only if this initial (and not “first”) case has a certain triviality that distinguishes it from the others
  • first ordinal number ; Ordinal number of the second kind and among these both the only finite and the only non-Limes number
  • smallest thickness of a set , at the same time the only one that already clearly defines the set (as the empty set )
  • only number where the sum with itself corresponds to the product with itself (this also applies to 2) and in addition the respective results are equal to the number itself.
  • smallest characteristic of a ring
  • Degree of constant polynomials (excluding the zero polynomial)

Until 1

• 0.0112359550561797 ... (Follow A021093 in OEIS )
  • ${\displaystyle {\tfrac {1}{89}}}$is the value of the infinite series , the summands of which are the product of the -th Fibonacci number with . ${\displaystyle \textstyle \sum _{n=1}^{\infty }f_{n}10^{-(n+1)}}$ ${\displaystyle n}$ ${\displaystyle f_{n}}$ ${\displaystyle 10^{-(n+1)}}$
• 0.12345678910111213141516 ... (sequence A033307 in OEIS )
  • ${\displaystyle C_{10}=\sum _{n=1}^{\infty }\sum _{k=10^{n-1}}^{10^{n}-1}{\frac {k}{10^{kn-9\sum _{j=0}^{n-1}10^{j}(n-j-1)}}}}$: The Champernowne number is the first constructed normal number .
• 0.2078795763507619 ... (Follow A049006 in OEIS )
  • ${\displaystyle {\mathrm {i} }^{\mathrm {i} }}$: The imaginary unit to the power has the real value (see also Euler's identity ). ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {i} }$ ${\displaystyle e^{-\pi /2}}$
• 0.2247448713915890 ... (see sequence A115754 in OEIS )
  • ${\displaystyle {\sqrt {\tfrac {3}{2}}}-1}$: Relative distance of the optimal support points from the edges of a uniformly loaded beam ( Bessel points ).
• 0.235711131719232931374143 ... (Follow A33308 in OEIS )
  • ${\displaystyle C_{10}=\sum _{n=1}^{\infty }p_{n}10^{-\left(n+\sum _{k=1}^{n}\lfloor \log _{10}{p_{k}}\rfloor \right)}}$: The Copeland-Erdős number is a normal number .
• 0.2614972128476427 ... (Follow A077761 in OEIS )
  • Meissel-Mertens constant M ₁ (prime number analogue to Euler-Mascheroni constant )
• 0.2801694990238691 ... (sequence A073001 in OEIS )
  • Bernstein constant β (the error of the best uniform approximation of | x | on [−1,1] by polynomials of even degree n is ~ β / n)
• 0.3036630028987326… (Follow A038517 in OEIS )
  • Gauss-Kusmin-Wirsing constant λ (occurs when describing the convergence of the number distribution in continued fraction expansion )
• 0.3532363718549959 ... (Follow A085849 in OEIS )
  • Hafner-Sarnak-McCurley constant (asymptotic probability that the determinants of two integer matrices are relatively prime) ${\displaystyle \sigma =\textstyle \prod \limits _{p\;{\text{prim}}}\!\!\!{\Bigl (}\!1\!-\!{\bigl (}1\!-\!\!\prod \limits _{k=1}^{\infty }(1\!-\!{\frac {1}{p^{k}}}\!){\bigr )}^{\!2}\!{\Bigr )}}$
• 0.3678794411714423 ... (Follow A068985 in OEIS )
  • Reciprocal of Euler's number ${\displaystyle e}$
  • Minimal digit of the function , since the zero is from and therefore also from . ${\displaystyle f(x)=x^{x}}$ ${\displaystyle 1/e}$ ${\displaystyle \ln(x)+1\ }$ ${\displaystyle f'(x)=x^{x}\cdot (\ln(x)+1)}$
• 0.4142135623730950 ... (sequence A014176 in OEIS )
  • ${\displaystyle \tan 22{,}5^{\circ }={\sqrt {2}}-1}$; algebraic value of the tangent function for half-integer argument in degrees
• 0.4342944819032518 ... (sequence A002285 in OEIS )
  • The logarithm of ten of Euler's number ${\displaystyle e}$
• 0.5
  • ${\displaystyle \sin {\tfrac {\pi }{6}}=\sin 30^{\circ }=\cos 60^{\circ }={\tfrac {1}{2}}}$; rational value of the sine and cosine functions
• 0.5432589653429767 ... (Follow A081760 in OEIS )
  • currently the most precise upper limit of the Landau constant (maximum, so that for every holomorphic function ƒ with ƒ ′ (0) = 1 there is a circular disk with a radius in the image of the unit disk ) ${\displaystyle {\mathfrak {L}}}$ ${\displaystyle {\mathfrak {L}}}$
• 0.5671432904097838 ... (Follow A019474 in OEIS )
  • The Ω constant : solution of the equation and thus the function value of the Lambert W function ${\displaystyle x=\exp(-x)}$ ${\displaystyle W(1)}$
• 0.5772156649015328 ... (sequence A001620 in OEIS )
  • Value of the Euler-Mascheroni constant , where denotes the harmonic series . ${\displaystyle \gamma =\lim _{n\to \infty }\left(H_{n}-\ln n\right)}$ ${\displaystyle H_{n}}$
• 0.5960631721178216 ... (Follow A051158 in OEIS )
  • Irrational value of the sum of the reciprocal of all Fermat numbers , that is ${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{F_{n}}}=\sum _{n=0}^{\infty }{\frac {1}{2^{2^{n}}+1}}}$
• 0.6180339887498948 ... (sequence A094214 in OEIS )
  • ${\displaystyle {\tfrac {{\sqrt {5}}-1}{2}}}$, i.e. reciprocal of the golden ratio and at the same time the golden ratio reduced by one: ${\displaystyle \Phi }$ ${\displaystyle 1/\Phi =\Phi -1}$
• 0.6243299885435508 ... (sequence A084945 in OEIS )
  • Golomb-Dickman constant ${\displaystyle \textstyle \lambda =\int _{0}^{1}e^{\operatorname {li} (x)}~dx}$
• 0.6309297535714574 ... (Follow A102525 in OEIS )
  • Hausdorff dimension of the fractal Cantor set , ${\displaystyle \log _{3}2}$
• 0.6434105462883380 ... (Follow A118227 in OEIS )
  • Cahen constant C ( transcendental number with a simple formation law for the part of the denominator of the fraction expansion )
• 0.6601618158468695 ... (sequence A005597 in OEIS )
  • Prime twin constant C ₂ (part of the Hardy-Littlewood conjecture about the number of prime twins ≤ x )
• 0.6627434193491815 ... (sequence A033259 in OEIS )
  • Limit of Laplace (maximum eccentricity for which the Laplace series converges to solve the Kepler equation )
• 0.6922006275553463 ... (sequence A072364 in OEIS )
  • value of ${\displaystyle (1/e)^{1/e}}$
  • global minimum of the function ${\displaystyle f(x)=x^{x}}$
• 0.6931471805599453 ... (sequence A002162 in OEIS )
  • Value of the logarithm naturalis of , i.e. value of ${\displaystyle 2}$ ${\displaystyle \textstyle \ln(2)=\sum _{k=1}^{\infty }{\tfrac {(-1)^{k+1}}{k}}=1-{\tfrac {1}{2}}+{\tfrac {1}{3}}-{\tfrac {1}{4}}+{\tfrac {1}{5}}-\dotsb }$
• 0.70258 ... (Follow A118288 in OEIS )
  • Embree-Trefethen constant β * (limit coefficient of generalized random Fibonacci sequences )
• 0.7071067811865475 ... (Follow A010503 in OEIS )
  • ${\displaystyle {\tfrac {1}{2}}{\sqrt {2}}}$, i.e. half of the root of 2 and at the same time its reciprocal
  • Value of the sine and cosine at , so ${\displaystyle 45^{\circ }}$ ${\displaystyle {\tfrac {1}{2}}{\sqrt {2}}=\cos 45^{\circ }=\sin 45^{\circ }}$
• 0.7390851332151606 ... (sequence A003957 in OEIS )
  • Fixed point of the cosine function , i.e. solution of the equation ${\displaystyle \cos(x)=x}$
• 0.7642236535892206 ... (Follow A064533 in OEIS )
  • Landau-Ramanujan constant
  • This results from the asymptotic determination of the proportion of the total of all natural numbers that can be represented as the sum of two square numbers.
• 0.8079455065990344 ... (Follow A133741 in OEIS )
  • Distance between the centers of two unit circles, each of which overlaps half of its area
• 0.8093940205406391… (Follow A085291 in OEIS )
  • Alladi-Grinstead constant (in n ! As the product of n prime powers, the largest possible smallest factor grows logarithmically ~  α  ln n ) ${\displaystyle \alpha =\textstyle \exp {\Bigl (}\!{\Bigl (}\sum \limits _{k=2}^{\infty }{\frac {1}{k}}\ln {\frac {k}{k-1}}{\Bigr )}-1{\Bigr )}}$ 
• 0.8660254037844386 ... (sequence A010527 in OEIS )
  • ${\displaystyle {\tfrac {1}{2}}{\sqrt {3}}}$, i.e. half of the root of 3
  • Value of the cosine at or the sine at ; so ${\displaystyle 30^{\circ }}$ ${\displaystyle 60^{\circ }}$ ${\displaystyle {\tfrac {1}{2}}{\sqrt {3}}=\cos 30^{\circ }=\sin 60^{\circ }}$
• 0.87058838 ... (sequence A213007 in OEIS )
  • Brun's constant ; Sum of the reciprocal values ​​of all prime quadruplets ${\displaystyle B_{4}}$
• 0.9159655941772190 ... (sequence A006752 in OEIS )
  • Catalan's constant ; ${\displaystyle G={\tfrac {1}{1^{2}}}-{\tfrac {1}{3^{2}}}+{\tfrac {1}{5^{2}}}-{\tfrac {1}{7^{2}}}+{\tfrac {1}{9^{2}}}-\dotsb }$
  • Functional value of the Dirichlet beta function ${\displaystyle \beta (2)}$
• 1
  • neutral element of multiplication in the ring of integers as well as its extension rings (these include the fields of rational , real and complex numbers ).
  • thus also the value of the empty product
  • formerly the first of the natural numbers
  • smallest positive integer
  • first index of countably indexed rows, as far as this is not used here (without exception is used for components of vectors and matrices) ${\displaystyle 0}$ ${\displaystyle 1}$
  • ${\displaystyle 1\cdot 1=1^{1}=1=1!}$only number in the product itself, the number itself and with itself the potency Faculty match; smallest of the two numbers for which the former two conditions or the latter two conditions apply
  • only Fibonacci number occurring more than once (namely twice) ; once (as the second of three) with their own index, once (as the first of three) smaller than their index (this is exactly one larger in all these cases), furthermore (as the first of four) with the distance from exactly to a prime number and which (as the second of four) is a non-first power ${\displaystyle 1}$
  • Definitions often demanded as the smallest thickness of a set for various applications, for example the smallest order of a ring (and, if an exception is not expressly inserted into the commonly formulated definition, also of a group )
  • smallest characteristic of a finite ring
  • first ordinal number of the first kind (successor number)
  • first Catalan number

Until 10

• 1.0149416064096536 ... (Follow A143298 in OEIS )
  • Gieseking's constant (maximum volume of a hyperbolic tetrahedron ) ${\displaystyle G=\textstyle \int _{0}^{2\pi /3}\ln(2\cos(x/2))\,\mathrm {d} x}$
• 1.0173430619844491 ... (sequence A013664 in OEIS )
  • Functional value of the zeta function ${\displaystyle \zeta (6)=\pi ^{6}/945}$
• 1.0594630943592952645618252949463 (12th root of 2)
  • Factor between the frequencies of two neighboring semitones (e.g. C and C #) with equal pitch
• 1.0823232337111381 ... (sequence A013662 in OEIS )
  • Functional value of the zeta function ${\displaystyle \zeta (4)=\pi ^{4}/90}$
• 1.08366 (sequence A228211 in OEIS )
  • Legendre constant (occurs in the prime number theorem , turned out to be incorrect)
• 1.0986858055251870 ... (Follow A086053 in OEIS )
  • Lengyel's constant Λ (occurs in the asymptotic analysis of the number of chains from the smallest to the largest element in the lattice of partitions )
• 1.1319882487943 ... (Follow A078416 in OEIS )
  • Viswanath constant K (basis of the asymptotically exponential growth of random Fibonacci sequences )
• 1.1547005383792515 ... (sequence A020832 in OEIS )
  • ${\displaystyle 1/\sin 60^{\circ }}$, Ratio of the circumferential radius to the incircular radius of the regular hexagon, determines the width of the hexagon socket wrench
• 1.1865691104156254 ... (sequence A100199 in OEIS )
  • Chintschin-Lévy constant ( almost everywhere the limit value for n → ∞ from ( ln q _n ) / n, where q _{n is} the denominator of the nth approximate fraction ) ${\displaystyle m=\pi ^{2}/(12\,\ln 2)}$
• 1.2020569031595942 ... (Follow A002117 in OEIS )
  • Functional value of the zeta function , Apéry constant ${\displaystyle \zeta (3)}$
• 1.2618595071429148 ... (Follow A100831 in OEIS )
  • Hausdorff dimension of the fractal Koch curve , ${\displaystyle \log _{3}4}$
• 1.2824271291006226 ... (Follow A074962 in OEIS )
  • Glaisher-Kinkelin constant A (occurs when evaluating integrals and series sums )
• 1.3063778838630806 ... (Follow A051021 in OEIS )
  • Mills' constant A (smallest number A> 0, so that for each n = 1, 2, 3, ... is a prime number , provided the Riemann hypothesis is correct) ${\displaystyle \lfloor A^{3^{n}}\rfloor }$
• 1.3247179572447460 ... (sequence A060006 in OEIS )
  • Plastic number (the unique real solution of the cubic equation ) ${\displaystyle x^{3}-x-1=0}$
• 1.4142135623730950 ... (sequence A002193 in OEIS )
  • ${\displaystyle {\sqrt {2}}}$, d. H. the square root of ( root of 2 ) ${\displaystyle 2}$
  • ${\displaystyle {\sqrt {2}}=2\sin 45^{\circ }=2\cos 45^{\circ }}$
  • Value of the length of the diagonal of a square with the length of the side ${\displaystyle 1}$
• 1.4513692348833810 ... (Follow A070769 in OEIS )
  • Ramanujan-Soldner constant , the only positive zero of the integral logarithm
• 1.4560749485826896 ... (Follow A072508 in OEIS )
  • Backhouse constant B (−1 / B is the zero of the power series with 1 and the prime numbers as coefficients)
• 1.4670780794339754 ... (Follow A086237 in OEIS )
  • Porter's constant (occurs in formulas of the asymptotic mean number of divisions in the Euclidean algorithm ) ${\displaystyle C={\tfrac {6\ln 2}{\pi ^{2}}}\left(3\ln 2+4\gamma -{\tfrac {24}{\pi ^{2}}}\zeta '(2)-2\right)-{\tfrac {1}{2}}}$
• 1.5849625007211561… (Follow A020857 in OEIS )
  • Hausdorff dimension of the fractal Sierpinski triangle , ${\displaystyle \log _{2}3}$
• 1.6066951524152917 ... (sequence A065442 in OEIS )
  • Erdős-Borwein constant (sum of the reciprocal values ​​of all Mersenne numbers )
• 1.6180339887498948 ... (sequence A001622 in OEIS )
  • Golden cut ${\displaystyle \Phi ={\tfrac {1+{\sqrt {5}}}{2}}}$
• 1.6449340668482264 ... (Follow A013661 in OEIS )
  • Functional value of the zeta function ${\displaystyle \zeta (2)=\pi ^{2}/6}$
• 1.7052111401053677 ... (Follow A033150 in OEIS )
  • Niven constant ( limit of the arithmetic mean of the maximum exponents of the prime factorization of the first natural numbers for ) ${\displaystyle n}$ ${\displaystyle n\to \infty }$
• 1.7320508075688772 ... (Follow A002194 in OEIS )
  • ${\displaystyle {\sqrt {3}}}$, the root of ( root of 3 ) ${\displaystyle 3}$
  • ${\displaystyle {\sqrt {3}}=2\sin 60^{\circ }=2\cos 30^{\circ }}$
  • Value of the length of the space diagonal of a cube with the side length ${\displaystyle 1}$
• 1.7724538509055160 ... (Follow A002161 in OEIS )
  • ${\displaystyle {\sqrt {\pi }}}$, the root of the circle number (root ) ${\displaystyle \pi }$
  • Function value of the gamma function ${\displaystyle \Gamma \left({\tfrac {1}{2}}\right)}$
  • Value of the error integral ${\displaystyle \textstyle \int \limits _{-\infty }^{\infty }e^{-x^{2}}\mathrm {d} x}$
• 1.851937052 ... (Follow A036792 in OEIS )
  • Wilbraham – Gibbs constant ${\displaystyle \int _{0}^{\pi }{\frac {\sin t}{t}}\ \mathrm {d} t=(1{,}851937052\dots )={\frac {\pi }{2}}+\pi \cdot (0{,}089490\dots )}$
• 1.90216058 ... (Follow A065421 in OEIS )
  • Brun's constant (sum of the reciprocal values ​​of all prime twins ) ${\displaystyle B_{2}}$
• 2
  • Smallest positive even number , defining for the even numbers
  • Least prime number
  • Single even prime number
  • Only number that is an odd phi function Euler has yet not to himself prime is
  • Smallest order of a body (required by definition)
  • Smallest characteristic of a finite body
  • Second Catalan number
  • The smallest basis of a value system , the dual system
  • ${\displaystyle 2+2=2\cdot 2=2^{2}}$. Hence, the only number where the sum with itself, the product with itself, and the power with itself (and the largest of only two if only the first two or only the last two conditions are required) is ${\displaystyle 2}$
  • ${\displaystyle 2=2!}$ Greatest number of two that matches its own faculty
  • Second of three Fibonacci numbers that are one smaller than their index, second of four that are exactly a prime number apart ${\displaystyle 1}$
  • The only natural number for which the equation is nontrivial and nevertheless solvable ( Fermat-Wiles theorem ) ${\displaystyle n}$ ${\displaystyle a^{n}+b^{n}=c^{n}}$
• 2.3025850929940456… (Follow A002392 in OEIS )
  • Logarithm naturalis of , i.e. value of ${\displaystyle 10}$ ${\displaystyle \ln 10={\tfrac {1}{\lg {e}}}}$
• 2.4142135623730950 ... (Follow A014176 in OEIS )
  • ${\displaystyle \tan {\tfrac {3\pi }{8}}=\tan 67{,}5^{\circ }={\sqrt {2}}+1}$ algebraic value of the tangent function
  • ${\displaystyle \delta _{S}=\lim _{n\to \infty }{\tfrac {F(n)}{F(n-1)}}={\sqrt {2}}+1}$ Silver cut , limit of the ratio of two consecutive numbers in the Pell sequence
• 2.5029078750958928 ... (Follow A006891 in OEIS )
  • One of the two Feigenbaum constants
• 2.5849817595792532 ... (Follow A062089 in OEIS )
  • Sierpiński constant K (occurs when estimating special sums)
• 2.6220575542921198 ... (Follow A062539 in OEIS )
  • Lemniscatic constant , defined as the value of the elliptic integral ${\displaystyle \varpi }$
• 2.6651441426902251 ... (Follow A007507 in OEIS )
  • The Gelfond-Schneider constant from the Gelfond-Schneider theorem
  • value of ${\displaystyle 2^{\sqrt {2}}}$
• 2.6854520010653064 ... (sequence A002210 in OEIS )
  • Khinchin's constant , almost everywhere , the geometric mean of the denominators of the continued fraction expansion
• 2.7182818284590452 ... (sequence A001113 in OEIS )
  • Euler's number , base of the natural logarithm ${\displaystyle e}$
• 2.8077702420285193… (Follow A058655 in OEIS )
  • Fransén-Robinson constant (area between the x -axis and the curve 1 / Γ ( x ) for x  > 0) ${\displaystyle F=\textstyle \int _{0}^{\infty }\!{\frac {1}{\Gamma (x)}}\,\mathrm {d} x}$
• 3
  • Least odd prime number
  • Fermat number ${\displaystyle F_{0}}$
  • Mersenne prime number ${\displaystyle M_{2}}$
  • Smallest natural number that is not a function value of the Euler φ function occurs
  • Largest Fibonacci number (of three) that is less than its index ( ); third of four Fibonacci numbers that are precisely spaced from a prime number ; only Fibonacci prime whose index is not prime ${\displaystyle 4}$ ${\displaystyle 1}$
• 3.1415926535897932384626433832795 ... (sequence A000796 in OEIS )
  • Circle number , ratio of the circumference of a circle to its diameter ${\displaystyle \pi }$
• 3.1428571428571428571428571428571 ... (Follow A068028 in OEIS )
  • ${\displaystyle {\tfrac {22}{7}}}$, Approximation to the circle number as it is often used ${\displaystyle \pi }$
• 3.3598856662431775531720113029189… (Follow A079586 in OEIS )
  • ' Fibonacci reciprocal constant', sum of the reciprocal values ​​of all Fibonacci numbers
• 4th
  • Number of corners of the regular polygon, the area of ​​which corresponds exactly to the second power of the edge length, which is why the term square defines both regular quadrilateral and second power
  • Smallest composite number
  • Number of colors sufficient to color any flat map ( four-color theorem )
  • ${\displaystyle 4=2+2=2\cdot 2=2^{2}}$
  • First non-Fibonacci number
  • Smallest Smith number
  • Number of faces and corners of a tetrahedron
  • Smallest natural number for which every nonnegative integer can be represented as a sum of at most square numbers (see: Waring's problem ) ${\displaystyle n}$ ${\displaystyle n}$
  • Number of points of the smallest affine plane
  • Smallest order of a non-commutative ring without a single element
  • Maximum degree of the general algebraic equation that can be solved using the extraction of roots
  • Smallest order of a non-cyclic group (the Klein group of four )
  • Minimum order of a body which is not a residue field is
• 4.6692016091029906 ... (sequence A006890 in OEIS )
  • Feigenbaum constant : Fixed point of the logistic equation , transition into chaos
• 5
  • Number of platonic solids
  • Smallest positive natural number , the square of which can be written as the sum of two positive square numbers : (see also: Pythagorean triple ) ${\displaystyle 5^{2}=3^{2}+4^{2}}$
  • Fermat number ${\displaystyle F_{1}}$
  • Largest number of corners of a regular polygon that appears as the side surface of a Platonic solid
  • The only component of two prime twins , namely and ${\displaystyle (3;5)}$ ${\displaystyle (5;7)}$
  • Least Wilson prime
  • Smallest possible mirp number , in the system of three the decimal is the same , the decimal is the same ${\displaystyle 5}$ ${\displaystyle 12}$ ${\displaystyle 7}$ ${\displaystyle 21}$
  • Third Catalan number
  • Largest (third) Fibonacci number that is identical to its own index
  • Smallest number for which a polygram exists
  • Number of corners of a polygon that has the same number of diagonals (a corner has diagonals) ${\displaystyle n}$ ${\displaystyle ((n-1)\cdot (n-2))/2-1}$
• 6th
  • Smallest perfect number : it is equal to the sum of their positive divisor other than their own: . ${\displaystyle 6=1+2+3}$
  • The number is equal to the product of its real divisors: ${\displaystyle 6=2\cdot 3}$
  • Exact common quotient of the areas of a regular hexagon and triangle for which the same side length is specified
  • Number of faces of the cube
  • Number of corners of the octahedron
  • Number of edges of the tetrahedron
  • In the plane, a circle can be touched by a maximum of other circles of the same size so that no overlaps occur. ${\displaystyle 6}$
  • Smallest positive integer whose cube can be written as the sum of three positive cubes: . ${\displaystyle 6^{3}=3^{3}+4^{3}+5^{3}}$
  • Largest order to which no Greco-Latin square exists
  • Smallest order of a non-Abelian group , the symmetric group ${\displaystyle S_{3}}$
  • Smallest positive natural number that is not a prime power
  • Smallest natural number greater than for which no field of order exists ${\displaystyle n}$ ${\displaystyle 1}$ ${\displaystyle n}$
  • Smallest, primarily pseudo-perfect number
  • Number of platonic solids in four dimensions
  • The only natural number above for which no connected polygram exists ${\displaystyle 4}$
• 6.283185307179586 ... (Follow A019692 in OEIS )
  • ${\displaystyle 2\pi }$: Scope of the unit circle
• 7th
  • Smallest number of vertices of a regular polygon that is not constructible with ruler and compass is
  • Mersenne prime number ${\displaystyle M_{3}}$
  • Number of colors that, according to the Ringel-Youngs Theorem, are sufficient to color any map on a torus
  • Smallest non-negative integer that cannot be written as a sum of less than four square numbers (see: Waring's problem )
  • Number of points and lines of the smallest projective plane , the Fano plane
  • Smallest positive natural number for which rectangles of different positive edge lengths exist in pairs, which can be combined to form a rectangle ${\displaystyle n}$ ${\displaystyle n}$
• 8th
  • Number of faces of the octahedron and number of corners of the cube
  • third of four Fibonacci numbers that are non-first powers and besides the trivial and single cube number; If you consider the Fibonacci number , the value is obtained by raising the value to the power of its index, the index ( ), when multiplying both numbers; is also the largest of four Fibonacci numbers , which are exactly the distance from a prime number ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle F_{3}=2}$ ${\displaystyle 8}$ ${\displaystyle 6}$ ${\displaystyle 8}$ ${\displaystyle 1}$
  • Least order of a non-commutative unitary ring
  • The only number with four divisors, the second largest of which is even.
• 9
  • Every positive natural number that is multiplied by gives the final number after the formation of cross-sums of the intermediate results . Examples: or . ${\displaystyle 9}$ ${\displaystyle 9}$ ${\displaystyle 8\cdot 9=72\rightarrow 7+2=9}$ ${\displaystyle 22\cdot 9=198\rightarrow 1+9+8=18\rightarrow 1+8=9}$
  • If you take any three-digit number in which the first and last digit differ by at least and take the same number with the reverse order of digits and form the difference between the two numbers, you get a multiple of . If you add this number to the number that has the reverse sequence of digits, you get the number . ${\displaystyle 2}$ ${\displaystyle 9}$ ${\displaystyle 1089=9\cdot 11^{2}}$
  • Smallest odd composite number
  • Smallest natural number for which every nonnegative integer can be represented as the sum of at most positive cubic numbers (see: Waring's problem ) ${\displaystyle n}$ ${\displaystyle n}$
  • Smallest positive natural number for which there are pairs of squares with different positive edge lengths that can be combined to form a rectangle ${\displaystyle n}$ ${\displaystyle n}$
  • Smallest order of a non-Desarguean projective plane
• 10
  • Largest number of corners of a regular polygon that appears as the side of an Archimedean solid
  • Smallest natural number for which applies to all natural numbers ( is Euler's φ function .) ${\displaystyle a}$ ${\displaystyle n-\varphi (n)\neq a}$ ${\displaystyle n}$ ${\displaystyle \varphi }$
  • Is also used as an approximation for . ${\displaystyle \pi ^{2}}$

Until 100

• 11
  • Length of the Golay code , the only nontrivial perfect ternary code that can correct more than one error. ${\displaystyle G_{11}}$
  • Smallest prime that is not a Mersenne prime . ${\displaystyle p}$ ${\displaystyle 2^{p}-1}$
  • Smallest repunit prime
• 12
  • Least abundant number .
  • Number of faces of the dodecahedron , number of edges of the cube and octahedron , number of corners of the icosahedron .
  • 1st sublime number and only one under a trillion
  • 3. Pentagonal number
  • 4. Number of rectangles
  • Three-dimensional kiss number
  • Order of the rotating group of the tetrahedron , the alternating group . ${\displaystyle A_{4}}$
  • A dozen
• 13
  • Number of Archimedean solids, if no distinction is made between similar solids.
  • 2. Wilson prime number .
  • Smallest number in the decimal system
• 14th
  • Number of three-dimensional Bravais grids
  • Smallest even natural number that does not appear as a function value of Euler's φ-function .
  • Fourth Catalan number .
• 14,134725141734693 ... (Follow A058303 in OEIS )
  • Imaginary part of the absolute smallest nontrivial zero of the zeta function ${\displaystyle {\tfrac {1}{2}}\pm \sigma i}$
• 15th
  • Number of Archimedean solids if non-reflection invariant solids are counted twice.
  • Smallest composite number for which, apart from isomorphism, only a single group of the order exists. ${\displaystyle n}$ ${\displaystyle n}$
  • Smallest pseudoprime number . Smallest natural number that cannot be written as a sum of less than eight cube numbers (see: Waring's problem ).
  • Largest binary value that a 4-bit variable can assume: ${\displaystyle 15=2^{4}-1=[1111]_{2}=[F]_{16}}$
  • Smallest natural number that Euler's φ-function has no prime number in common.
• 16
  • ${\displaystyle 16=2^{4}=4^{2}}$; is actually the only number for which mutually distinct natural numbers and exist with . ${\displaystyle 16}$ ${\displaystyle n}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle n=a^{b}=b^{a}}$
  • Smallest natural number , so that with a finite number of exceptions, every natural number can be written as a sum of at most biquadrates (see: Waring's problem ). ${\displaystyle n}$ ${\displaystyle n}$
  • Order of the smallest unitary ring that is not anti-isomorphic to itself.
  • Number of binary values ​​that a 4-bit variable can accept: ${\displaystyle 16=2^{4}}$
• 17th
  • Fermat number . ${\displaystyle F_{2}}$
  • Number of crystallographic groups in the plane .
  • Gauss considered the construction of the regular 17-sided with compass and ruler to be one of his most important discoveries.
• 18th
  • The first maximum of the number of non-isomorphic cubic cage graphs of a given waist size , which is reached with increasing waist size of these graphs at . ${\displaystyle \nu }$ ${\displaystyle \nu =9}$
  • The only number that is double its checksum.
  • Smallest number with six divisors, which are always alternately odd and even, sorted by size.
• 19th
  • Smallest natural number for which every positive natural number can be represented as a sum of at most biquadrates (see: Waring's problem ). ${\displaystyle n}$ ${\displaystyle n}$
  • Largest non-square integer for which the ring is Euclidean . ${\displaystyle d}$ ${\displaystyle \mathbb {Z} [{\sqrt {d}}]}$
• 20th
  • Number of faces of the icosahedron and number of corners of the dodecahedron .
  • “ God's number ” of the Rubik's cube : maximum number of turns that are necessary to solve a Rubik's cube from any position
  • Least abundant number without a perfect divisor
• 21st
  • Smallest positive natural number for which there are pairs of squares with different positive edge lengths that can be put together to form a square. ${\displaystyle n}$ ${\displaystyle n}$
• 22nd
  • The first coefficient of the continued fraction representation of . ${\displaystyle \pi ^{e}}$
• 23
  • Smallest positive natural number for which cuboids with different positive edge lengths exist, which can be combined to form a cuboid. ${\displaystyle n}$ ${\displaystyle n}$
  • Smallest and next to the only natural number that cannot be written as a sum of less than nine cubic numbers (see Waring's problem ). ${\displaystyle 239}$
  • Length of the Golay code , the only nontrivial perfect binary code that can correct more than one error. ${\displaystyle G_{23}}$
  • smallest prime number outside of a prime number twin (if one disregards the one whose distance to neighboring prime numbers is even closer than provided in the definition of the prime number twin) ${\displaystyle 2}$
• 24
  • Order of the rotation group symmetrical group of the cube and the octahedron . ${\displaystyle S_{4}}$
  • Largest natural number with the property that all natural numbers are less than a divisor of . ${\displaystyle n}$ ${\displaystyle {\sqrt {n}}}$ ${\displaystyle n}$
• 25th
  • Least square number, the sum of two square numbers is: ${\displaystyle 3^{2}+4^{2}=5^{2}=25}$
  • Smallest natural number with a multiplicative tenacity of . ${\displaystyle 2}$
• 26th
  • Number of sporadic groups
  • The only natural number that has a square and a cube number as neighbors
• 27
  • The smallest natural number that can be written as the sum of three square numbers in two different ways, namely as . ${\displaystyle 3^{2}+3^{2}+3^{2}=5^{2}+1^{2}+1^{2}}$
  • The number of lines on a projective cubic surface .
• 28
  • The smallest natural number that can be written as the sum of four square numbers in two different ways, namely as . ${\displaystyle 4^{2}+2^{2}+2^{2}+2^{2}=5^{2}+1^{2}+1^{2}+1^{2}}$
  • Second perfect number .
• 29
  • Least prime number, which is the sum of three consecutive square numbers: ${\displaystyle 2^{2}+3^{2}+4^{2}=29}$
• 30th
  • Number of edges of the dodecahedron and the icosahedron .
  • Area number of the rhombic triacontahedron . Smallest Giuga number .
  • The largest natural number with the property that of other than all natural numbers smaller than that to be prime, prime numbers are. ${\displaystyle n}$ ${\displaystyle 1}$ ${\displaystyle n}$ ${\displaystyle n}$
• 31
  • Mersenne prime number ${\displaystyle M_{5}}$
• 32
  • Number of crystal classes in the three-dimensional crystal lattice
• 33
  • The largest natural number that cannot be represented as the sum of different triangular numbers. ${\displaystyle n}$
• 34
  • The smallest number that has the same number of divisors as its predecessor and successor.
• 35
  • The smallest tetrahedral number that is the product of a prime twin .
• 36
  • First (non-trivial) square triangular number , a triangular number at the same time perfect square is.
  • The only (non-trivial) triangular number whose square root ( ) is also a triangular number: ${\displaystyle 6}$ ${\displaystyle \Delta _{8}=(\Delta _{3})^{2}}$
• 37
  • Smallest natural number for which every nonnegative whole number can be represented as the sum of at most the fifth powers of nonnegative whole numbers (see: Waring's problem ). ${\displaystyle n}$ ${\displaystyle n}$
  • Least irregular prime number .
  • It is the fourth number .
• 38
  • The row sum of the only nontrivial magic hexagon with the side length . ${\displaystyle n=3}$
• 39
  • Smallest natural number with a multiplicative tenacity of . ${\displaystyle 3}$
• 40
  • Smallest natural number not in this list .
• 41
  • The polynomial yields for for all prime numbers. ${\displaystyle n^{2}+n+a}$ ${\displaystyle a=41}$ ${\displaystyle n\in \{0,\dotsc ,a-2\}}$
• 42
  • Second primarily pseudo-perfect number .
  • Fifth Catalan number .
  • Smallest-dimensional space without a sausage catastrophe .
• 43
  • Largest natural number for which it is impossible to put together Chicken McNuggets in the usual packs of 6, 9 and 20 (see coin problem ). ${\displaystyle n}$ ${\displaystyle n}$
• 44
  • Number of possibilities to solve the house of St. Nicholas ; another 44 variants are reflections of these paths
• 49
  • Smallest pseudo-prime number (number that composed , even though you can see it either on the final number yet on the cross sum)
• 50
  • Smallest natural number that can be written in two different ways as the sum of two square numbers: ${\displaystyle 50=5^{2}+5^{2}=7^{2}+1^{2}}$
• 56
  • At this number, the sausage disaster occurs .
• 60
  • The order of the alternating group , i.e. the smallest non- resolvable group and the smallest non- Abelian simple group . ${\displaystyle A_{5}}$
  • Number of vertices of four Archimedean solids : the truncated dodecahedron , the truncated icosahedron or football body, the small rhombicosidodecahedron and the beveled dodecahedron (Dodekaedron simum).
  • Number of edges of two Archimedean solids : the icosidodecahedron and the beveled cube (Cubus simus).
  • The smallest natural number that is divided by all natural numbers up to . ${\displaystyle 5}$
  • The smallest natural number that is divided by all natural numbers up to . ${\displaystyle 6}$
• 65
  • Smallest natural number that can be written in two different ways as the sum of two different square numbers: ${\displaystyle 65=1^{2}+8^{2}=4^{2}+7^{2}}$
• 70
  • Smallest odd number
• 71
  • Largest supersingular prime . Largest right-trimmable prime number to the base . ${\displaystyle 3}$
• 72
  • Smallest positive integer whose fifth power can be written as the sum of five fifth powers of positive natural numbers: . ${\displaystyle 72^{5}=19^{5}+43^{5}+46^{5}+47^{5}+67^{5}}$
• 73
  • It is the 21st prime number, is the product of and . ${\displaystyle 21}$ ${\displaystyle 7}$ ${\displaystyle 3}$
  • Its mirror number is the 12th prime number (again mirror number of ). ${\displaystyle 37}$ ${\displaystyle 21}$
  • In binary notation it is a palindromic number : . The palindrome has seven digits and contains three times the . ${\displaystyle 1001001}$ ${\displaystyle 1}$
  • In octal, there is a palindromic number: . The palindrome has three digits and contains three times the . ${\displaystyle 111}$ ${\displaystyle 1}$
  • It is the sixth mirp number .
• 77
  • Smallest natural number with a multiplicative tenacity of . ${\displaystyle 4}$
• 79
  • Smallest natural number that cannot be written as the sum of fewer than biquadrates (see: Waring's problem ). ${\displaystyle 19}$
• 80
  • “ God's number ” for the 15 puzzle : maximum number of moves that are necessary to solve the puzzle from any position
• 85
  • 85 can be represented in two different ways as the sum of two square numbers: ${\displaystyle 85=9^{2}+2^{2}=7^{2}+6^{2}}$
• 88
  • Number of ways to draw the house of Nicholas , see number ${\displaystyle 44}$
• 92
  • Number of Johnson bodies
  • Number of faces of the beveled dodecahedron .
  • Number of solutions to the 8 queens problem

Up to 1000

• 101
  • The smallest three-digit prime number
  • First three-digit number palindrome
• 105
  • The circular division polynomial is the first whose coefficients are not all , or . ${\displaystyle \Phi _{105}}$ ${\displaystyle -1}$ ${\displaystyle 0}$ ${\displaystyle 1}$
• 107
  • Smallest three-digit number .
• 108
  • Regular pentagon angle
• 109.47 ...
  • Tetrahedron angle
  • ${\displaystyle \arccos \left(-{\tfrac {1}{3}}\right)}$
• 111
  • Third smallest repunit number
• 120
  • Maximum number of corners of an Archimedean solid in the large rhombicosidodecahedron
• 127
  • Mersenne prime number . ${\displaystyle M_{7}}$ ${\displaystyle 2^{7}-1}$
• 132
  • Sixth Catalan number .
• 143
  • Solution of Waring's problem for k = 7
• 144
  • Smallest positive integer whose fifth power can be written as the sum of four fifth powers of positive natural numbers: . This identity was discovered in 1966 and refuted a generalization of Fermat's great theorem suggested by Leonhard Euler in 1769 . ${\displaystyle 144^{5}=27^{5}+84^{5}+110^{5}+133^{5}}$
  • Largest and fourth fibonacci number (after , and ) that is a non-first power, including the only non-trivial square number. It is also the square of its own Fibonacci index. ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 8}$
• 153
  • You start with any natural number divisible by three and continuously build the sum of the cubes of the decimal digits: this sequence will always reach 153 and because of 1³ + 5³ + 3³ = 1 + 125 + 27 = 153 then it will stop there.
• 163
  • Largest number for which class number has. Therefore is unusually close to an integer. ${\displaystyle d}$ ${\displaystyle \mathbb {Q} ({\sqrt {-d}})}$ ${\displaystyle 1}$ ${\displaystyle \mathrm {e} ^{\pi {\sqrt {163}}}\approx 262537412640768743{,}99999999999925}$
• 168
  • Order of the second smallest nonabelian simple group.
• 180
  • Maximum number of edges of an Archimedean solid in the large rhombicosidodecahedron
• 191
  • Largest right-trimmable prime number to the base . ${\displaystyle 4}$
• 196
  • Smallest and best-known candidate for a Lychrel number .
• 210
  • Largest Goldbach number .
• 219
  • Number of three-dimensional symmetry groups without taking into account the orientation in space ( space group ).
• 220
  • Smallest friendly number , together with the smallest friendly number pair. ${\displaystyle 284}$
• 223
  • The only natural number that cannot be written as the sum of less than positive fifth powers (see: Waring's problem ). ${\displaystyle 37}$
• 230
  • Number of three-dimensional symmetry groups taking into account the orientation in space ( space group ).
• 239
  • The largest and the only natural number that cannot be written as the sum of less than nine cubic numbers (see: Waring's problem ). ${\displaystyle 23}$
• 248
  • Dimension of the complex Lie group . ${\displaystyle E_{8}}$
• 251
  • Smallest natural number that can be written as the sum of three cube numbers in two different ways, namely as ${\displaystyle 1^{3}+5^{3}+5^{3}=2^{3}+3^{3}+6^{3}}$
• 255
  • Largest binary value that an 8-bit variable can assume: ${\displaystyle 255=2^{8}-1=[1111.1111]_{2}=[FF]_{16}}$
• 256
  • Number of binary values ​​that an 8-bit variable can accept: ${\displaystyle 256=2^{8}}$
• 257
  • Fermat number . ${\displaystyle F_{3}}$
• 261
  • Number of three-dimensional networks in a four-dimensional cube.
• 284
  • Second smallest friendly number , together with the smallest friendly number pair. ${\displaystyle 220}$
• 292
  • Fifth number in the continued fraction expansion of the circle number . Since this number is relatively large, the continued fraction terminated after the fourth digit provides a very good approximation for : The two numbers match in six decimal places, which is a much better approximation than would be expected for an approximate fraction with a denominator of this magnitude. ${\displaystyle \pi }$ ${\displaystyle {\tfrac {355}{113}}}$ ${\displaystyle \pi }$
• 325
  • Smallest number that can be written in three ways as the sum of two square numbers: ${\displaystyle 325=1^{2}+18^{2}=6^{2}+17^{2}=10^{2}+15^{2}}$
• 341
  • Smallest pseudoprime to the base ${\displaystyle 2}$
• 353
  • Smallest positive natural number whose biquadrate can be written as the sum of four positive biquadrates: ${\displaystyle 353^{4}=30^{4}+120^{4}+272^{4}+315^{4}}$
• 373
  • The only three-digit number for which the following applies: The digits , and are prime numbers. The numbers and are prime numbers. The number is a prime number. (Special case of prime numbers that can be truncated on both sides ) ${\displaystyle ABC}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle AB}$ ${\displaystyle BC}$ ${\displaystyle ABC}$
• 420
  • The smallest natural number that is divided by all numbers from to . ${\displaystyle 2}$ ${\displaystyle 7}$
• 429
  • Seventh Catalan number .
• 454
  • Largest natural number that cannot be written as the sum of less than eight cubic numbers (see: Waring's problem ).
• 466
  • Largest natural number that cannot be written as the sum of fewer than positive integer fifths. (see: Waring's problem ). ${\displaystyle 32}$
• 495
  • Three- digit Kaprekar constant
• 496
  • Third perfect number
• 561
  • Smallest Carmichael number
• 563
  • Third and currently largest known Wilson prime
• 666
  • The sum of the squares of the first seven prime numbers
  • Is represented in Roman numerals as DCLXVI. Here each numerical value occurs exactly once in the order of decreasing size. ${\displaystyle 1000}$
  • The sum of the numbers from to ${\displaystyle 1}$ ${\displaystyle 36}$
  • See also six hundred and sixty-six
• 679
  • Smallest natural number with a multiplicative tenacity of . ${\displaystyle 5}$
• 840
  • The smallest natural number that is divided by all numbers from to . ${\displaystyle 2}$ ${\displaystyle 8}$
• 858
  • Second smallest Giuga number with four factors.
• 880
  • Number of fourth order magic squares that do not emerge from one another through reflection or rotation.
• 945
  • Least odd abundant number .
• 991
  • Largest known permutable prime number that is not a success .

Up to 10,000

• 1009
  • Smallest four-digit number
• 1089
  • For a three-digit number that is not a number palindrome , you create its mirror number (e.g. is the mirror number of ) and subtract the smaller number from the larger number; The reverse number of the result is then added to the result (if the first intermediate result only has two digits, the number is preceded by a zero); this method always gives the result ${\displaystyle 327}$ ${\displaystyle 723}$ ${\displaystyle 1089}$
• 1093
  • First Wieferich prime number
• 1105
  • Smallest number that can be written in four ways as the sum of two square numbers: ${\displaystyle 1105=4^{2}+33^{2}=9^{2}+32^{2}=12^{2}+31^{2}=23^{2}+24^{2}}$
• 1233
  • ${\displaystyle 12^{2}+33^{2}}$
• 1444
  • In the decimal system, square numbers can not end in more than three identical ( different) digits. is the smallest square number that has this maximum number of identical digits at the end. ${\displaystyle 0}$ ${\displaystyle 1444=38^{2}}$
• 1722
  • Third Giuga number .
• 1729
  • Smallest number that can be represented in two different ways as the sum of two third powers: ( Hardy - Ramanujan number). ${\displaystyle 10^{3}+9^{3}=12^{3}+1^{3}}$
  • The first Carmichael number of the form . ${\displaystyle (6n+1)\cdot (12n+1)\cdot (18n+1)}$
• 1806
  • Third, primarily pseudo-perfect number .
• 2047
  • ${\displaystyle M_{11}=2^{11}-1}$: the smallest Mersenne number with prime exponents that is not prime, i.e. not a Mersenne prime number: ${\displaystyle 2047=23\cdot 89}$
• 2437
  • Largest right-trimmable prime number to the base . ${\displaystyle 5}$
• 2520
  • The smallest natural number that is divided by all numbers from to . ${\displaystyle 2}$ ${\displaystyle 10}$
  • Eighteenth composite number - it has total divisors. In addition, it is the largest "special" highly composed number: The number of divisors is only exceeded when the numerical value is doubled ( has divisors). ${\displaystyle 48}$ ${\displaystyle 5040}$ ${\displaystyle 60}$
• 3003
  • The only known number so far that appears exactly eight times in Pascal's triangle . See Singmaster conjecture .
• 3435
  • First non-trivial Münchhausen number as a base , in which the sum of the individual digits, taken even up, results in the original number: ${\displaystyle 10}$ ${\displaystyle 3^{3}+4^{4}+3^{3}+5^{5}=27+256+27+3125=3435}$
• 3511
  • Second (and largest known) Wieferich prime number
• 4711
  • Is used as a metasyntactic variable for finitely large cardinal numbers; the background is that this figure just does not have special mathematical properties, but a well-known brand name for colognes is
• 5525
  • Smallest number that can be written in exactly six ways as the sum of two square numbers: ${\displaystyle 5525=7^{2}+74^{2}=14^{2}+73^{2}=22^{2}+71^{2}=25^{2}+70^{2}=41^{2}+62^{2}=50^{2}+55^{2}}$
• 5777 and 5993
  • the only two known odd numbers greater than that cannot be written as , where is a prime number and an integer ${\displaystyle 1}$ ${\displaystyle p+2\cdot n^{2}}$ ${\displaystyle p}$ ${\displaystyle n}$
• 6174
  • Kaprekar constant for four-digit numbers.
• 6788
  • Smallest natural number with a multiplicative tenacity of . ${\displaystyle 6}$
• 6841
  • Largest right-trimmable prime number to the base . ${\displaystyle 7}$
• 7825
  • Smallest number for which there is no binary coloring of the set up to without a single-colored Pythagorean triple . ${\displaystyle n}$ ${\displaystyle n}$
• 8125
  • Smallest number that can be written in exactly five ways as the sum of two square numbers: ${\displaystyle 8125=5^{2}+90^{2}=27^{2}+86^{2}=30^{2}+85^{2}=50^{2}+75^{2}=58^{2}+69^{2}}$
• 8128
  • Fourth perfect number
• 8191
  • Mersenne prime number ${\displaystyle M_{13}}$
• 8833
  • ${\displaystyle 88^{2}+33^{2}}$

Up to 1 million

• 10.100
  • ${\displaystyle 10^{2}+100^{2}}$(applies to all place value systems )
• 16,843
  • First Wolstenholme prime
• 27,720
  • The smallest natural number that is divided by all natural numbers up to . ${\displaystyle 11}$
  • The smallest natural number that is divided by all natural numbers up to . ${\displaystyle 12}$
• 29,341
  • 10. Carmichael number , smallest pseudo-prime to the bases , , , and . ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 5}$ ${\displaystyle 7}$ ${\displaystyle 11}$
• 41,041
  • Smallest Carmichael number with four prime factors
• 47,058
  • Fourth primarily pseudo perfect number .
• 63,973
  • Carmichael number ${\displaystyle M_{4}(1)}$
• 65,533
  • Functional value of the Ackermann function . ${\displaystyle a(4,1)=a(5,0)}$
• 65,535
  • Largest binary value that a 16-bit variable can assume: ${\displaystyle 65.535=2^{16}-1=[1111.1111.1111.1111]_{2}=[FFFF]_{16}}$
• 65,536
  • Number of binary values ​​that a 16-bit variable can accept: ${\displaystyle 65.536=2^{16}}$
• 65,537
  • Fermat number , largest known (and probably also largest) Fermat prime number ${\displaystyle F_{4}}$
• 66.198
  • Fourth Giuga number .
• 68,889
  • Smallest natural number with a multiplicative tenacity of . ${\displaystyle 7}$
• 78,557
  • Smallest known Sierpiński number .
• 108,863
  • Largest right-trimmable prime number to the base . ${\displaystyle 6}$
• 131,071
  • Mersenne prime number ${\displaystyle M_{17}}$
• 142,857
  • Smallest non-trivial cyclic number .
• 148,349
  • The only number that is equal to the sum of its sub-faculty digits.
• 177.147
  • Number of possibilities ( ) in the football pool (penalty bet). ${\displaystyle 3^{11}}$
• 271,441
  • The smallest Perrinsche pseudo prime number , . ${\displaystyle 521^{2}}$
• 294,409
  • Carmichael number ${\displaystyle M_{3}(6)}$
• 360.360
  • The smallest natural number that is divided by all natural numbers up to . ${\displaystyle 13}$
  • The smallest natural number that is divided by all natural numbers up to . ${\displaystyle 14}$
  • The smallest natural number that is divided by all natural numbers up to . ${\displaystyle 15}$
• 509.203
  • Smallest known trickle number .
• 524.287
  • Mersenne prime number ${\displaystyle M_{19}}$
• 549.945
  • 1. Kaprekar constant for six-digit numbers.
• 617.716
  • The -th triangular number , a palindrome of numbers ; Discovered by Charles Trigg . ${\displaystyle 1111}$
• 631.764
  • 2. Kaprekar constant for six-digit numbers.
• 720.720
  • The smallest natural number that is divided by all natural numbers up to . ${\displaystyle 16}$
• 990.100
  • ${\displaystyle 990^{2}+100^{2}}$

Up to 1 billion

• 2,082,925
  • Smallest number that can be written in different ways as the sum of two square numbers : ${\displaystyle 18}$

  ${\displaystyle {\begin{aligned}2.082.925&=26^{2}+1443^{2}=134^{2}+1437^{2}=163^{2}+1434^{2}=195^{2}+1430^{2}=330^{2}+1405^{2}=370^{2}+1395^{2}=429^{2}+1378^{2}=531^{2}+1342^{2}=541^{2}+1338^{2}=\\&=558^{2}+1331^{2}=579^{2}+1322^{2}=702^{2}+1261^{2}=730^{2}+1245^{2}=755^{2}+1230^{2}=845^{2}+1170^{2}=894^{2}+1133^{2}=926^{2}+1107^{2}=1014^{2}+1027^{2}\end{aligned}}}$

• 2,124,679
  • Second Wolstenholme prime number
• 2,677,889
  • Smallest natural number with a multiplicative tenacity of . ${\displaystyle 8}$
• 4,005,625
  • Smallest number that can be written in ways as the sum of two square numbers ${\displaystyle 20}$
• 4,497,359
  • Largest right-trimmable prime number to the base . ${\displaystyle 8}$
• 5,882,353
  • ${\displaystyle 588^{2}+2353^{2}}$
• 5,928,325
  • Smallest number that can be written in ways as the sum of two square numbers ${\displaystyle 24}$
• 9,721,368
  • Largest number made up of different digits (in the decimal system) from which any digit can be crossed out so that the rest can be divided by the crossed out digit
• 26,888,999
  • Smallest natural number with a multiplicative tenacity of . ${\displaystyle 9}$
• 33,550,336
  • Fifth perfect number
• 56.052.361
  • Carmichael number ${\displaystyle M_{3}(35)}$
• 73.939.133
  • Largest right-trimmable prime number in the decimal system : For the number, when the last digit is deleted, a prime number with precisely this property is created; ie , , , , , , also prime numbers. ${\displaystyle 7393913}$ ${\displaystyle 739391}$ ${\displaystyle 73939}$ ${\displaystyle 7393}$ ${\displaystyle 739}$ ${\displaystyle 73}$ ${\displaystyle 7}$
• 87,539,319
  • Smallest number that can be represented in three different ways as the sum of two cubic numbers: Taxicab number ${\displaystyle \operatorname {Ta} (3)}$
• 94.122.353
  • ${\displaystyle 9412^{2}+2353^{2}}$
• 118.901.521
  • Carmichael number ${\displaystyle M_{3}(45)}$
• 146.511.208
  • Narcissistic Number  : ${\displaystyle 1^{9}+4^{9}+6^{9}+5^{9}+1^{9}+1^{9}+2^{9}+0^{9}+8^{9}}$
• 172,947,529
  • Carmichael number ${\displaystyle M_{3}(51)}$
• 216.821.881
  • Carmichael number ${\displaystyle M_{3}(55)}$
• 228.842.209
  • Carmichael number ${\displaystyle M_{3}(56)}$
• 275.305.224
  • Number of fifth order magic squares that do not emerge from one another by mirroring or rotating.
• 472.335.975
  • Narcissistic number ${\displaystyle 4^{9}+7^{9}+2^{9}+3^{9}+3^{9}+5^{9}+9^{9}+7^{9}+5^{9}}$
• 534.494.836
  • Narcissistic number ${\displaystyle 5^{9}+3^{9}+4^{9}+4^{9}+9^{9}+4^{9}+8^{9}+3^{9}+6^{9}}$
• 635.318.657
  • Smallest number that can be written in two different ways as the sum of two biquadrates, namely as . ${\displaystyle 158^{4}+59^{4}=133^{4}+134^{4}}$
• 906.150.257
  • Smallest counterexample to the conjecture of Pólya
• 912.985.153
  • Narcissistic number ${\displaystyle 9^{9}+1^{9}+2^{9}+9^{9}+8^{9}+5^{9}+1^{9}+5^{9}+3^{9}}$

Up to 1 trillion

• 1,299,963,601
  • Carmichael number ${\displaystyle M_{3}(100)}$
• 1,355,840,309
  • Largest right-trimmable prime number to the base . ${\displaystyle 9}$
• 1,765,038,125
  • ${\displaystyle 17650^{2}+38125^{2}}$
• 2,147,483,647
  • Mersenne prime number ${\displaystyle M_{31}}$
• 2,214,408,306
  • Fifth Giuga number .
• 2,214,502,422
  • Fifth primarily pseudo perfect number .
• 2,301,745,249
  • Carmichael number ${\displaystyle M_{3}(121)}$
• 2,584,043,776
  • ${\displaystyle 25840^{2}+43776^{2}}$
• 3,778,888,999
  • Smallest natural number with a multiplicative tenacity of . ${\displaystyle 10}$
• 3,816,547,290
  • The only pandigital number whose first digits (read as numbers) are divisible by: the first digit by , the first two digits by , the first three digits by , etc. ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$
• 4,294,967,295
  • Largest value that can be represented as an unsigned 32-bit integer : ${\displaystyle 4.294.967.295=2^{32}-1=[1111.1111.1111.1111.1111.1111.1111.1111]_{2}=[FFFF.FFFF]_{16}}$
• 4,294,967,296
  • Number of binary values ​​that a 32-bit variable can accept: ${\displaystyle 4.294.967.296=2^{32}}$
• 4,294,967,297
  • Using this number, Euler refuted a conjecture made by Fermat - see Fermat's prime number .
• 4,679,307,774
  • Narcissistic number ${\displaystyle 4^{10}+6^{10}+7^{10}+9^{10}+3^{10}+0^{10}+7^{10}+7^{10}+7^{10}+4^{10}}$
• 5,391,411,025
  • Smallest abundant number that is neither divisible by nor by . ${\displaystyle 2}$ ${\displaystyle 3}$
• 6,172,882,716
  • The -th triangular number , a palindrome of numbers . Discovered by Charles Trigg . ${\displaystyle 111.111}$
• 7,416,043,776
  • ${\displaystyle 74160^{2}+43776^{2}}$
• 8.235.038.125
  • ${\displaystyle 82350^{2}+38125^{2}}$
• 8,589,869,056
  • Sixth perfect number , discovered by Cataldi in 1588 .
• 15.170.835.645
  • Smallest number that can be written in three different ways as the sum of two cube numbers each , namely as ${\displaystyle 517^{3}+2468^{3}=709^{3}+2456^{3}=1733^{3}+2152^{3}}$
• 24,423,128,562
  • Sixth Giuga number .
• 32.164.049.650
  • Narcissistic number ${\displaystyle 3^{11}+2^{11}+1^{11}+6^{11}+4^{11}+0^{11}+4^{11}+9^{11}+6^{11}+5^{11}+0^{11}}$
• 52.495.396.602
  • Sixth primarily pseudo perfect number .
• 116.788.321.168
  • ${\displaystyle 116788^{2}+321168^{2}}$
• 123.288.328.768
  • ${\displaystyle 123288^{2}+328768^{2}}$
• 137,438,691,328
  • Seventh perfect number , discovered by Cataldi in 1588 .
• 192.739.365.541
  • Carmichael number ${\displaystyle M_{4}(45)}$
• 200.560.490.131
  • Is the prime number , where is the product of all prime numbers from to (see also Euclid's theorem , prime faculty ). ${\displaystyle 31\#+1}$ ${\displaystyle 31\#}$ ${\displaystyle 2}$ ${\displaystyle 31}$
• 461.574.735.553
  • Carmichael number ${\displaystyle M_{4}(56)}$
• 876.712.328.768
  • ${\displaystyle 876712^{2}+328768^{2}}$
• 883.212.321.168
  • ${\displaystyle 883212^{2}+321168^{2}}$

Up to 1 trillion

• 7,625,597,484,987
  • ${\displaystyle 3^{3^{3}}=3^{27}=7\,625\,597\,484\,987{\color {red}\neq }\left(3^{3}\right)^{3}=27^{3}=19\,683}$
• 10,028,704,049,893
  • Carmichael number ${\displaystyle M_{4}(121)}$
• 28,116,440,335,967
  • ${\displaystyle 2^{14}+8^{14}+1^{14}+1^{14}+6^{14}+4^{14}+4^{14}+0^{14}+3^{14}+3^{14}+5^{14}+9^{14}+6^{14}+7^{14}}$
• 61,728,399,382,716
  • The -th triangular number , a palindrome of numbers . ${\displaystyle 11.111.111}$
• 277,777,788,888,899
  • Smallest natural number with a multiplicative tenacity of . ${\displaystyle 11}$
• 432.749.205.173.838
  • The seventh Giuga number
• 4,338,281,769,391,370
  • ${\displaystyle 4^{16}+3^{16}+3^{16}+8^{16}+2^{16}+8^{16}+1^{16}+7^{16}+6^{16}+9^{16}+3^{16}+9^{16}+1^{16}+3^{16}+7^{16}+0^{16}}$
• 9,585,921,133,193,329
  • The smallest Carmichael number according to the Richard GE Pinch system
• 14.737.133.470.010.574
  • The eighth Giuga number
• 21,897,142,587,612,075
  • ${\displaystyle 2^{17}+1^{17}+8^{17}+9^{17}+7^{17}+1^{17}+4^{17}+2^{17}+5^{17}+8^{17}+7^{17}+6^{17}+1^{17}+2^{17}+0^{17}+7^{17}+5^{17}}$
• 48,988,659,276,962,496
  • The smallest number that can be written in five different ways as the sum of two cubic numbers each, namely as ${\displaystyle 231.518^{3}+331.954^{3}=221.424^{3}+336.588^{3}=205.292^{3}+342.952^{3}=107.839^{3}+362.753^{3}=38.787^{3}+365.757^{3}}$
• 262,537,412,640,768,743.9999999999992500 ... (Follow A060295 in OEIS )
  • ${\displaystyle e^{\pi \cdot {\sqrt {163}}}\approx 640320^{3}+744}$is called Ramanujan's constant , is a transcendent number and is very close to an integer.
• 550.843.391.309.130.318
  • The ninth Giuga number

Over 1 trillion

• 1,517,841,543,307,505,039
  • ${\displaystyle 1^{19}+5^{19}+1^{19}+7^{19}+8^{19}+4^{19}+1^{19}+5^{19}+4^{19}+3^{19}+3^{19}+0^{19}+7^{19}+5^{19}+0^{19}+5^{19}+0^{19}+3^{19}+9^{19}}$
• 2,305,843,008,139,952,128
  • The eighth perfect number , discovered by Leonhard Euler in 1750 .
• 2,305,843,009,213,693,951
  • Mersenne prime number ${\displaystyle M_{61}}$
• 12,157,692,622,039,623,539
  • ${\displaystyle 1^{1}+2^{2}+1^{3}+5^{4}+7^{5}+6^{6}+9^{7}+2^{8}+6^{9}+2^{10}+2^{11}+0^{12}+3^{13}+9^{14}+6^{15}+2^{16}+3^{17}+5^{18}+3^{19}+9^{20}}$
• 18,446,744,073,709,551,615
  • Largest binary value that a 64-bit variable can assume: ${\displaystyle 18.446.744.073.709.551.615=2^{64}-1=[1111.1111.1111.1111.1111.1111.1111.1111.1111.1111.1111.1111.1111.1111.1111.1111]_{2}=[FFFF.FFFF.FFFF.FFFF]_{16}}$
• 18,446,744,073,709,551,616
  • Number of binary values ​​that a 64-bit variable can accept: ${\displaystyle 18.446.744.073.709.551.616=2^{64}}$
• 63,105,425,988,599,693,916
  • ${\displaystyle 6^{20}+3^{20}+1^{20}+0^{20}+5^{20}+4^{20}+2^{20}+5^{20}+9^{20}+8^{20}+8^{20}+5^{20}+9^{20}+9^{20}+6^{20}+9^{20}+3^{20}+9^{20}+1^{20}+6^{20}}$
• 128,468,643,043,731,391,252
  • ${\displaystyle 1^{21}+2^{21}+8^{21}+4^{21}+6^{21}+8^{21}+6^{21}+4^{21}+3^{21}+0^{21}+4^{21}+3^{21}+7^{21}+3^{21}+1^{21}+3^{21}+9^{21}+1^{21}+2^{21}+5^{21}+2^{21}}$
• 357,686,312,646,216,567,629,137
  • Largest prime number that can be truncated to the left in the decimal system: If you remove any part of the number from the front (left), a prime number always remains.
• 244,197,000,982,499,715,087,866,346
  • The tenth known Giuga number
• 618,970,019,642,690,137,449,562,111
  • Mersenne prime number ${\displaystyle M_{89}}$
• 554.079.914.617.070.801.288.578.559.178
  • The eleventh known Giuga number .
• 8,490,421,583,559,688,410,706,771,261,086
  • The seventh primarily pseudo-perfect number .
• 162.259.276.829.213.363.391.578.010.288.127
  • Mersenne prime number ${\displaystyle M_{107}}$
• 1,910,667,181,420,507,984,555,759,916,338,506
  • The twelfth known Giuga number .
• 2,658,455,991,569,831,744,654,692,615,953,842,176
  • The ninth perfect number , discovered by Pervusin in 1883 .
• 170,141,183,460,469,231,731,687,303,715,884,105,727
  • Mersenne prime number ${\displaystyle M_{127}}$
• 191,561,942,608,236,107,294,793,378,084,303,638,130,997,321,548,169,216
  • The tenth perfect number , discovered by Ralph E. Powers in 1911 .
• 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
  • The order of the monster group (the largest sporadic group ).
• 13,164,036,458,569,648,337,239,753,460,458,722,910,223,472,318,386,943,117,783,728,128
  • The eleventh perfect number , discovered by Ralph E. Powers in 1914 .
• 6,086,555,670,238,378,989,670,371,734,243,169,622,657,830,773,351,885,970,528,324,860,512,791,691,264
  • The second raised number , discovered by Kevin Brown
• 14.474.011.154.664.524.427.946.373.126.085.988.481.573.677.491.474.835.889.066.354.349.131.199.152.128
  • The twelfth perfect number , discovered by Lucas in 1876 .
• 2 ⁵²⁰ (2 ⁵²¹ - 1)
  • The 13th perfect number , discovered by Raphael M. Robinson in 1952 .
• 2 ⁶⁰⁶ (2 ⁶⁰⁷ - 1)
  • The 14th perfect number , discovered by Raphael M. Robinson in 1952 .
• 2 ¹²⁷⁸ (2 ¹²⁷⁹ - 1)
  • The 15th perfect number , discovered by Raphael M. Robinson in 1952 .
• 2 ²²⁰² (2 ²²⁰³ - 1)
  • The 16th perfect number , discovered by Raphael M. Robinson in 1952 .
• 2 ²²⁸⁰ (2 ²²⁸¹ - 1)
  • The 17th perfect number , discovered by Raphael M. Robinson in 1952 .
• 1.29 x 10 ⁸⁶⁵
  • The lower bound for the maximum number of ones in a holding Busy Beaver with six states
• 2 ³²¹⁶ (2 ³²¹⁷ - 1)
  • The 18th perfect number , discovered by Riesel in 1957 .
• 3 × 10 ¹⁷³⁰
  • The lower bound for the maximum number of steps a holding Busy Beaver with six states can take
• 2 ⁴²⁵² (2 ⁴²⁵³ - 1)
  • The 19th perfect number , discovered by Adolf Hurwitz and Selfridge in 1961 .
• 2 ⁴⁴²² (2 ⁴⁴²³ - 1)
  • The 20th perfect number , discovered by Adolf Hurwitz and Selfridge in 1961 .
• 2 ⁹⁶⁸⁸ (2 ⁹⁶⁸⁹ - 1)
  • The 21st perfect number , discovered by Gillies in 1963 .
• 2 ⁹⁹⁴⁰ (2 ⁹⁹⁴¹ - 1)
  • The 22nd perfect number , discovered by Gillies in 1963 .
• 2 ^11,212 (2 ^11,213 - 1)
  • The 23rd perfect number , discovered by Gillies in 1963 .
• 2 ^19,936 (2 ^19,937 - 1)
  • The 24th perfect number , discovered by Tuckermann in 1971 .
• 2 ^21,700 (2 ^21,701 - 1)
  • The 25th perfect number , discovered by Noll and Nickel in 1978 .
• 2 ^23,208 (2 ^23,209 - 1)
  • The 26th perfect number , discovered by Noll in 1979 .
• 2 ^65,536 - 3rd
  • Function value of the Ackermann function (decimal number with digits) ${\displaystyle a(4,2)}$ ${\displaystyle 19.729}$
• 2 ^44,496 (2 ^44,497 - 1)
  • The 27th perfect number , discovered by Slowinski and Nelson in 1979 .
• 2 ^86,242 (2 ^86,243 - 1)
  • The 28th perfect number , discovered by Slowinski in 1982 .
• 48,047,305,725 × 2 ^172,403 - 1
  • Largest known Sophie Germain prime up to 2008 .
• 2 ^110.502 (2 ^110.503 - 1)
  • The 29th perfect number , discovered by Colquitt and Welsh in 1988 .
• 2 ^{132 048} (2 ^{132 049} - 1)
  • The 30th perfect number , discovered by Slowinski in 1983 .
• 2 ^{216 090} (2 ^{216 091} - 1)
  • The 31st perfect number , discovered by Slowinski in 1985 .
• 481,899 × 2 ^481,899 + 1
  • Largest known Cullen prime until 2008 ${\displaystyle C_{481.899}}$
• 2 ^{756 838} (2 ^{756 839} - 1)
  • The 32nd perfect number , discovered by Slowinski and Gage in 1992 .
• 2 ^{859 432} (2 ^{859 433} - 1)
  • The 33rd perfect number , discovered by Slowinski in 1993 .
• 3,752,948 × 2 ^3,752,948 - 1
  • The largest known Woodall prime ${\displaystyle W_{3.752.948}}$
• 6,679,881 × 2 ^6,679,881 + 1
  • The largest known Cullen prime ${\displaystyle C_{6.679.881}}$
• 2 ^25,964,951 - 1
  • The 42nd well-known Mersenne prime , a number with digits ${\displaystyle 7.816.230}$
• 2 ^30,402,457 - 1
  • The 43rd known Mersenne prime , a number with digits ${\displaystyle 9.152.052}$
• 2 ^32,582,657 - 1
  • The 44th known Mersenne prime , a number with digits ${\displaystyle 9.808.358}$
• 2 ^37,156,667 - 1st
  • The 45th known Mersenne prime , a number with digits ${\displaystyle 11.185.272}$
• 2 ^42,643,801 - 1
  • The 46th known Mersenne prime , a number with digits ${\displaystyle 12.837.064}$
• 2 ^43,112,609 - 1
  • The 47th known Mersenne prime , a number with digits ${\displaystyle 12.978.189}$
• 70388830… 50240001
  • The largest Carmichael number found (up to 1996) that has different prime divisors. It was found by Löh and Niebuhr, a number with digits ${\displaystyle 1.101.518}$ ${\displaystyle 16.142.049}$
• 2 ^57,885,161 - 1
  • The 48th known Mersenne prime , a number with digits ${\displaystyle 17.425.170}$
• 2 ^74.207.281 - 1
  • The 49th known Mersenne prime , a number with digits ${\displaystyle 22.338.618}$
• 2 ^74.207.280 (2 ^74.207.281 - 1)
  • The 50th known Mersenne prime , a number with digits ${\displaystyle 23.249.425}$
• 2 ^82,589,933 - 1
  • The 51st known Mersenne prime number and thus the largest known prime number (as of December 7, 2018), a number with digits ${\displaystyle 24.862.048}$
• ${\displaystyle 9^{9^{9}}=9^{387420489}\approx 4{,}281\ldots \cdot 10^{369693099}}$
  • Largest number with digits that can be written with three decimal digits ${\displaystyle 369.693.100}$
• 2 ^{2,305,843,009,213,693,951} - 1
  • This double Mersenne number , which can also be written as and has around 694 quadrillion digits, is possibly a prime number. Refuting this is the declared task of the GIMPS project , which coordinates distributed computing power over the Internet. ${\displaystyle 2^{2^{61}-1}-1}$
• ${\displaystyle 2^{2^{5.523.858}}+1\approx 10^{10^{1662846,4284}}}$
  • ${\displaystyle F_{5.523.858}}$is the largest Fermat number so far (as of January 31, 2020) for which a prime factor is known. It has more than jobs. If you were to write this number on a square piece of paper with 16 digits per cm², the square piece of paper would have an area of ​​approx. 10 ^1,662,809 square light years, ^i.e. a side length of approx. 10 ^831,404 light years . ${\displaystyle 10^{1.662.846}}$
• ${\displaystyle e^{e^{e^{79}}}\approx 10^{10^{10^{34}}}}$
  • Skewes number , for a long time (1931–1971) the largest finite number used in a mathematical proof. If you were to write this number on a square sheet of paper with 16 digits per cm², the square sheet of paper would have an area of ​​approximately square light years , i.e. a side length of approximately light years (the exponent therefore has 34 digits). ${\displaystyle 10^{8{,}852\cdot 10^{33}}}$ ${\displaystyle 10^{4{,}426\cdot 10^{33}}}$
• Mega
  • In Steinhaus-Moser notation :
• Megiston
  • In Steinhaus-Moser notation :
• Moser's number
  • Also of Steinhaus and Moser named
• Graham's number () ${\displaystyle G_{64}}$
  • Ousted Skewes' number from number one in the largest finite numbers used in a mathematical proof.

Infinite sizes

• ${\displaystyle \infty }$
  • Infinity, the reciprocal of 0 in certain computing systems , is greater than all numbers in this list and is not itself a number. With can indeed be expected to a limited extent, however, many expressions that contain either himself or (namely the terms and to the extent that no rule of de l'Hospital can be applied) is not defined. ${\displaystyle \infty }$ ${\displaystyle \infty }$ ${\displaystyle \infty }$ ${\displaystyle 0\cdot \infty }$ ${\displaystyle \infty /\infty }$
• ${\displaystyle -\infty }$
  • smaller than all ( whole , rational , real ) numbers, otherwise see above
  • in some geometries, but not on the usual number line, applies ${\displaystyle -\infty =\infty }$
  • the only negative and only infinite value that can appear as the degree of a polynomial (namely the zero polynomial ).
• ${\displaystyle \aleph _{0}}$( aleph ), (small omega ) ${\displaystyle 0}$ ${\displaystyle \omega }$
  • ${\displaystyle \aleph _{0}}$is the countable cardinality of the natural , rational and algebraic numbers and thus the smallest transfinite cardinal number . is the smallest ordinal number that is larger than any natural number, and thus the smallest transfinite ordinal number. It is true, but the arithmetic of the ordinal numbers is different from that of the cardinal numbers. ${\displaystyle \omega }$ ${\displaystyle \omega =\aleph _{0}}$
  • ${\displaystyle \omega }$is the second ordinal number of the second kind (i.e. number without a predecessor). All of these numbers are called Limes numbers , so it is their first. ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle \omega }$
• ${\displaystyle \epsilon _{0}}$
  • The smallest ordinal number that cannot be achieved with a finite number of arithmetic operations (addition, multiplication, exponentiation) . It is still countable , therefore . ${\displaystyle \omega }$ ${\displaystyle \omega <\varepsilon _{0}<\omega _{1}}$
• ${\displaystyle \epsilon _{1}}$
  • The smallest ordinal number that cannot be counted.
• ${\displaystyle \aleph _{1}}$
  • The next greater thickness, that is . If one accepts the continuum hypothesis, it agrees with the width of the continuum (the set of real numbers). ${\displaystyle \aleph _{0}}$ ${\displaystyle \omega _{1}=\aleph _{1}}$ ${\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}}$
• ${\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}}$
  • The uncountable power of the continuum, the irrational , transcendental , real and complex numbers and quaternions , the power of the power set of natural numbers.
• ${\displaystyle {\mathfrak {f}}=2^{\mathfrak {c}}}$
  • The uncountable power of the real functions.

Complex numbers

In this sub-list special complex numbers are collected and sorted according to their amount.

• i
  • The imaginary unit. A complex number whose square has the value and which is therefore the solution to the quadratic equation . is fourth root of unity . The formal definition is set (instead of what is also possible ). See also imaginary numbers . ${\displaystyle -1}$ ${\displaystyle x^{2}+1=0}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle {\mathrm {i} }=(0,1)\in \mathbb {R} ^{2}}$ ${\displaystyle (0,-1)}$
• −i
  • Reciprocal of the imaginary unit ${\displaystyle \mathrm {i} }$
  • ${\displaystyle {\mathrm {i} }\cdot (-{\mathrm {i} })=1}$or (inverse element of multiplication, but here also of addition:) . is like fourth root of unity . ${\displaystyle {\tfrac {1}{\mathrm {i} }}=-{\mathrm {i} }}$ ${\displaystyle {\mathrm {i} }+(-{\mathrm {i} })=0}$ ${\displaystyle -{\mathrm {i} }}$ ${\displaystyle \mathrm {i} }$
• ${\displaystyle {\tfrac {1}{2}}\cdot (-1\pm {\mathrm {i} }\cdot {\sqrt {3}})}$
  • The Primitive Third Roots of Unity ; is the third power of these two numbers . ${\displaystyle 1}$
• πi
  • Returns the value as the argument of the exponential function , see Euler's identity . ${\displaystyle -1}$
• 2πi
  • Period of the complex exponential function.
• 1/2 + i 14.134725141734693… (sequence A058303 in OEIS )
  • Zero of the Riemann zeta function with the smallest, positive imaginary part.

Numbers of particular importance

Until 0

• 0
  • The ice point describes the freezing point of water under normal conditions in degrees Celsius.
  • As absolute zero , 0 Kelvin, which corresponds to −273.15 ° C or −459.67 ° F, represents the theoretically lowest possible temperature, which, however, cannot be reached in practice
  • Network elimination number in many telephone networks (simply in area codes (D) / area codes (A) and mobile network codes, doubled in country codes)
  • Call of the switchboard in many private branch exchanges

Until 1

• 0.0078749969978123 (sequence A100264 in OEIS )
  • Chaitin's constant Ω (approximate, incalculable probability with which a universal Turing machine stops on any input)
• 0.5
  • As a fraction ½ (a half) the only real fraction that has always had a special name in most languages.
• 1
  • Numerical value of the Milesian - Greek number alpha .
  • Roman numeral I.
  • Symbol of unity and origin

Until 10

• 1.0594630943592952 ... (sequence A010774 in OEIS )
  • ${\displaystyle {\sqrt[{12}]{2}}}$, Frequency ratio between two adjacent semitones with equal pitch
• 1.2589254117941673 ... (sequence A011279 in OEIS )
  • ${\displaystyle {\sqrt[{10}]{10}}}$, Logarithmic comparative value 1 decibel (dB)
• 1.4
  • Popular approximation for , for example, the aperture range in photography : 1.0; 1.4; 2.0; 2.8; 4.0; 5.6; 8th; 11; 16; 22; ... ${\displaystyle {\sqrt {2}}}$
• 1.4142135623730950 ... (sequence A002193 in OEIS )
  • ${\displaystyle {\sqrt {2}}}$, Aspect ratio of many paper formats , for example DIN A and DIN B formats with the aspect ratio ${\displaystyle 1:{\sqrt {2}}}$
• 1.5
  • With the special designation “one and a half” traditionally linguistically particularly emphasized fractional number. Other languages ​​(e.g. Russian - полтора́) also have a special name for this number.
• 1.5396007178390020 ... (Follow A118273 in OEIS )
  • Lieb's ice cube constant (residual entropy of ice is N k ln W in an exactly solvable 2D model in statistical physics ) ${\displaystyle W=(4/3)^{3/2}}$   
• 2
  • Symbol of opposites.
  • In Chinese philosophy, yin and yang .
  • Numerical value of the Milesian- Greek number beta .
  • Code for (English) "to" in abbreviations , for example in B2C = Business-to-Consumer .
  • Number of points that define a straight line.
• 3
  • Number of repetitions for affirmation (affirmation) in mythology and spirituality .
  • "Number of all good things."
  • So many times Peter denied before a rooster crowed ( Passion ).
  • Trinity : Father, Son, Holy Spirit.
  • Three theological virtues : faith, hope, love.
  • Number of classic aggregate states .
  • Third gender for people who deviate from heteronormative rules.
  • Numerical value of the Milesian- Greek number gamma .
  • Number of points to define a plane.
  • three modern levels of the Catholic ordination office (also earlier main levels): bishop, priest and deacon
• 3.2
  • The old aperture series in photography is based on multiples of 3.2 (actually of ): 1.1, 1.6, 2.2, 3.2, 4.5, 6.3, 9, 12.5, 18, 25, 36, 50, 71, 100. ${\displaystyle {\sqrt {10}}}$
• 4th
  • Number of elements in ancient times.
  • Four directions . Four seasons .
  • Four canonical gospels and evangelists .
  • Four cardinal virtues .
  • Chinese and Japanese unlucky number (pronounced like "death").
  • In the western world it stands for luck (clover leaf).
  • Numerical value of the Milesian- Greek number Delta .
  • Minimum number of points to define a body
  • Language short code for (English) "for", for example in 4U = for you.
• 5
  • Number of elements in Asia, partly also in Greek mysticism ( Quintessenz , Aither )
  • Base number in ancient Egypt in the sense of ( pyramid ) and in multiples of 5, probably symbolic for the human body: five (four plus one) limbs, fingers, toes. ${\displaystyle 5=4+1}$
  • The pentagram (five-pointed star) is ascribed a magical peculiarity.
  • Numerical value of the mileso- Greek number epsilon .
  • Prescribed number of legs (possibly with castors) for office swivel chairs in order to avoid accidental tipping, as the contact radius around a (regular) pentagon no longer fluctuates as much as in a square.
  • Roman numeral V
  • Holy number among the Manichaeans
• 6th
  • Number of quarks (up, down, charm, strange, top and bottom).
  • The hexahedron (cube) is one of the platonic solids .
  • The Star of David , an example of a hexagram , is a hexagonal star.
  • Numerical value of the Milesian- Greek number stigma .
  • The symmetry of the snowflake is sixfold. Because of the special structure of the water molecules, only angles of 60 ° or 120 ° are possible.
• 7th
  • Number of days in a week .
  • Often used in fairy tales :
    • The wolf and the seven young goats
    • Snow White and the Seven Dwarfs
    • Thumbnail and the Seven Mile Boots
    • The seven Ravens
    • The seven Swabians
    • Seven beautiful
    • The brave little tailor killed "seven [flies] in one go"
  • Numerical value of the Milesian- Greek number zeta .
  • Number of crystal systems of the three-dimensional lattice.
  • In Christianity the 7 symbolizes completeness.
    • Number of days of the creation cycle
    • Noah was to take seven pairs of pure animals into the ark .
    • In Revelation : seven churches, seals , trumpets, angels, plagues, bowls, seven-headed beast.
    • In Catholicism 7 sacraments , 7 main virtues (three divine and four cardinal virtues), 7 root sins , (rarely) 7 heavenly virtues (counter-virtues to the root sins), 7 gifts of the Holy Spirit , 7 each spiritual and physical works of mercy , 7 classical levels of the Ordination office (without the superordinate and extraordinary episcopal ordination): Ostiarier, lecturer, exorcist, acolyte, subdeacon, deacon and priest (today limited to religious groups)
  • Number of celestial bodies visible to the naked eye that appear to be movable from the earth as the central point ( sun , moon , Mercury , Venus , Mars , Jupiter , Saturn )
  • Miller's number describes the fact established by George A. Miller that a person can only hold 7 ± 2 information units (chunks) in short-term memory at the same time
  • Cloud seven describes as a winged word a heavenly feeling of happiness in matters of love (in English Cloud 9)
• 8th
  • Lucky number in China
  • Holy number in India
  • Numerical value of the Milesian- Greek number Eta .
  • In our solar system , eight planets orbit the sun.
  • Language short code for the German syllable "Acht", z. B. "Good N8"
  • Language short code for the English syllable "ight / ite / ate", as in "good n8" or "2 L8"
  • in Christianity the number of supernatural abundance (compared to perfection 7): resurrection on the 8th day, 8 beatitudes
• 9
  • Numerical value of the Milesian- Greek number theta
  • holy number of the Baha'i
  • special number in Norse mythology , e.g. B. the number of nights that Odin hung wounded on the world tree and devised the runes
  • Number of the Self in Satanism
  • Emergency call channel in the CB radio on 27.065 MHz AM
  • Number of supposed lives of a cat
• 9.8066500 (Follow A072915 in OEIS )
  • (normalized) value of the acceleration due to gravity in m / s² - in practice often rounded to 9.81 or 9.8 or 10
• 10
  • Basis of our number system ( decimal system )
  • The Ten Commandments
  • Number of fingers and toes
  • The 10 is symbolic of earthly perfection
  • Numerical value of the Milesian- Greek number Iota
  • In Christianity and Judaism , the 10 symbolizes completeness (but inferior to the 7).
  • Roman numeral X

Until 100

• 11
  • Smallest number of schnapps
  • Foolish number in the Rhenish Carnival :
    • Beginning of the carnival on 11.11. at 11 a.m. 11
    • The Elferrat is the parliament of the foolish kingdom in Carnival, Mardi Gras and Mardi Gras
  • The "soccer eleven": each team has eleven players on the field
  • Formerly known as the "dirty dozen"
  • Number (next to 12), which is not pronounced in decimal but still according to a historical twelve system with "eleven"; the decimal formulation would be "oneteen"
• 12
  • Number of pentominos
  • A dozen
  • The basis of prehistoric payment systems
  • A symbol of perfection
  • In the Bible ...
    • the 12 tribes of Israel and often references to them
    • the 12 apostles of Jesus
  • 12 is the mean number of hours the sun shows itself during the day and the number of months of the year
  • In music, an octave consists of 12 semitones
  • There are 12 signs of the zodiac
  • 12 Olympic gods
  • King Eurystheus gave Heracles 12 tasks ("Dodekathlos")
  • 12 inhabited islands of the Dodecanese
  • 12 stars on the European flag
  • Number (next to 11), which is not pronounced in a decimal but still according to a historical twelve system with "twelve"; the decimal formulation would be "two thirteen"
  • According to the old German spelling, it is traditionally the last written number. Today you can also write smaller numbers in digits and larger numbers.
• 13
  • Unlucky number and / or lucky number
  • The Wild Thirteen
  • In German and in all Germanic languages ​​first composite number (e.g. thirteen in English ), the numbers 11 and 12 have their own names (e.g. in English eleven and twelve ).
• 14th
  • Number of stations on a way of the cross
  • Chinese unlucky number (pronounced like "certain death" (without escape))
  • Children's prayer "14 angels stand around me" (originally from Engelbert Humperdinck's opera Hansel and Gretel )
  • The fourteen emergency helpers
• 15th
  • 15 minutes stand for ¼ hour
  • Scoring for volleyball in the 5th and beach volleyball in the 3rd set (if there is at least 2 points difference to the opposing team)
• 16
  • At sixteen, in many societies, you reach a preliminary stage of adulthood, for example the age of consent in Switzerland or the driver's license in the USA
• 17th
  • Unlucky number in Italy
  • According to Kabbalistic number mysticism , 17 corresponds to the numerical value of the Hebrew word טוב ("good")
• 18th
  • The 18th birthday is the age of majority in most states
  • For the Jews, for whom numbers are expressed by letters, the numerical value means 18 lives
  • The Israelites had 18 minutes to leave Egypt
  • The matzos for the Passach festival may not be made for more than 18 minutes
  • Under neo-Nazis code number for "AH / Adolf Hitler", after the first and eighth letters of the alphabet
• 19th
  • In Islam, the entrance to hell is guarded by 19 angels
• 20th
  • Numerical value of the Milesian- Greek number kappa
• 21st
  • Number of points aimed for in the game of Black Jack or 17 and 4
  • Scoring for beach volleyball in the 1st and 2nd set (with at least 2 points difference to the opposing team)
  • Formerly the age of majority
  • Sum of the eyes of a dice
  • "Half the truth" as an allusion to the one described in The Hitchhiker's Guide to the Galaxy as an answer to the question about "life, the universe and all the rest" 42
• 22nd
  • Number of letters in the Hebrew alphabet
• 23
  • Plays a role in various conspiracy theories , u. a. as an alleged number of the Illuminati
  • Smallest number of people with random birthdays who are more likely to have two birthdays on the same day than to all have birthdays on different days ( birthday problem )
  • Humans ( homo sapiens ) have 23 pairs of chromosomes, with the 23rd pair also being the sex-specifying one.
• 24
  • Number of hours in a day
  • Number of books of the Tanakh
  • Number of letters in the Greek alphabet
• 25th
  • Anniversary number ; often referred to as the "Silver Jubilee ", e.g. B. " Silver Wedding " on the 25th wedding anniversary
• 26th
  • Number of letters in the Latin alphabet
• 27
  • Number of books of the New Testament
• 27,322:
  • The number of days it takes for the moon to orbit the earth ( sidereal month )
• 28
  • Under neo-Nazis code number for "Blood & Honor", after the second and eighth letters of the alphabet
  • 4 weeks have 28 days
  • Number of days in February in the "normal" calendar year
  • Number of letters in the Arabic alphabet
• 29
  • Number of days in February in the leap year
• 29.530588 ...
  • Days, synodic period of the moon (after that the moon phases are repeated)
• 30th
  • Number of days in April, June, September, and November
  • Numerical value of the Milesian- Greek number lambda
  • Number of days in a month for the interest calculation (in Germany )
• 31
  • Number of days in January, March, May, July, August, October, and December
• 32
  • Number of cards in Skat ("Skatblatt") and Schafkopf (version "long sheet")
• 36
  • Number of cards in Jassen
• 37
  • Number of numbers to bet on in French roulette
• 39
  • Number of books of the Old Testament in the German Protestant Bible editions
• 40
  • Stands as a symbol for testing, probation, initiation, death
  • Ali Baba and the 40 thieves
  • Minimum age of the Federal President in Germany
  • In the Bible ...
    • the (actual) flood lasted 40 days
    • was Isaac 40 years old, when he Rebekah took to wife
    • was Esau 40 years old, when he took to wife Judith
    • Moses was with God 40 days and 40 nights to receive the law
    • the Israelite exodus from Egypt lasted 40 years
    • was Joshua 40 when he was sent out by Moses the country " Kadesh Barnea to spy out"
    • was Ishbosheth 40 when he became king over Israel
    • King David ruled Israel for 40 years, King Joash also ruled for 40 years
    • Elijah fasted forty days and nights and went to Horeb during that time
    • fasted Jesus 40 days in the desert (hence the duration of the course, far easier fasting after the church) and was from the devil tried
    • the time between the resurrection and the ascension of Jesus lasted 40 days (hence the feast date)
  • The plague quarantine lasted 40 days
  • Number of cards in the Doppelkopf (version "without Luschen") and in an Ecuadorian card game ("Cuarenta" = German "Forty")
  • Numerical value of the Milesian- Greek number My
• 42
  • In the novel The Hitchhiker's Guide to the Galaxy , the number 42 appears as an answer to the question about "life, the universe and all the rest"
• 43
  • Atomic number of the first chemical element without stable isotopes ( technetium )
  • Spanish liquor Licor 43 (Cuarenta Y Tres)
• 46
  • Typical number of human chromosomes
  • Number of books of the (Catholic) Old Testament
  • according to the Bible ( Joh 2.20  EU ) the duration of the construction of the Herodian temple
  • Numerical value of the name Adam (occurs as an interpretation of the aforementioned Bible passage)
• 48
  • Number of cards in the Doppelkopf (version "with nines")
• 50
  • Anniversary number ; often referred to as the "Golden Jubilee ", e.g. B. “ Golden Wedding ” on the 50th wedding anniversary
  • The Jewish weekly festival Shavuot is celebrated 50 days, i.e. seven weeks plus one day after the Passover , based on Pentecost (Greek: Pentekoste) on the fiftieth day after Easter (including Easter)
  • Numerical value of the Milesian- Greek number Ny
  • Roman numeral L
• 52
  • Holy number of the Maya , after 52 years the calendar starts again
  • Number of cards in bridge , poker and blackjack
• 52.1775
  • Average number of weeks in a year taking into account leap years
• 53
  • Herbie's starting number in the film "A great Beetle" (VW)
  • Book title “53 Eine Claimung” (2009) by Thomas Trenkler, traces the number 53
• 55
  • Good luck, radio operator
• 60
  • One shock, five dozen
  • Highest score that can be achieved with a single throw while playing darts
  • Number of carbon atoms in the simplest fullerene C ₆₀
  • Numerical value of the Milesian- Greek number Xi
• 62
  • Number of months in a Yuga period
• 64
  • Number of hexagrams in Yijing
  • Number of fields on a chessboard
• 66
  • Number of books of the Bible in the German Protestant Bible editions
  • In the English-speaking world, the opening quotation marks (“) are sometimes jokingly called 66 due to their shape - analogous to 99 for the closing quotation marks (”)
  • for one of the first continuous road connections in the USA, Route 66
• 69
  • A sexual position in which both partners simultaneously satisfy each other orally
• 70
  • Numerical value of the Milesian- Greek number omicron
  • often simplifying for the number of peoples according to the Bible (actually 72)
• 72
  • In Islam, the number of huri (paradise virgins) with which some believers are rewarded after their death
  • Number of the peoples of the earth according to the Bible ( Gen 10  EU )
  • based on the earlier upper limit for the number of cardinals (obsolete)
• 73
  • Number of books in the Catholic Bible
  • Many greetings, radio operator code
• 75
  • Fax extension, (in Austria) frequently used telephone extension for fax connection in an office
• 80
  • Number of elements with at least one stable isotope
  • Numerical value of the Milesian- Greek number Pi
• 81
  • Tetragrams in the I-Ching = number of verses from Laotse's " Tao te king "
  • Abbreviation for the Hells Angels , since H is the eighth letter and A is the first letter of the alphabet
• 82
  • Atomic number of lead , the element with the highest atomic number, which is a stable isotope has
• 88
  • Literally: "No matter how ~"
  • Under neo-Nazis code number for "HH" / Heil Hitler, since H is the eighth letter of the alphabet
  • Radio language: "love and kisses"
  • In China, abbreviation for "Bye-Bye" because of the pronunciation of the numbers
• 90
  • Right angle , measured in degrees
  • Numerical value of the Milesian- Greek number Qoppa
• 97
  • Often chosen as an example for any number; many libraries stamp page 97
• 99
  • Last whole number before the hundred, is often used as a literary element in the sense of “one before completeness”, for example in Nena's 99 balloons, the song “99 bottles of beer” and 99 names of Allah
  • Number of months in an octahedral period
  • Get out of here, radio operator
• 100
  • Right angle , measured in gons
  • Boiling point of water in degrees Celsius under normal conditions
  • Numerical value of the Milesian- Greek number Rho
  • Roman numeral C

Up to 1000

• 101
  • Room 101 appears in several novels and films, for example in the novel 1984 by George Orwell , Matrix , A Beautiful Mind , Kill Bill - Volume 2 etc.
  • Taipei 101 - nickname of the Taipei Financial Center (Chinese 台北 101, Táiběi yīlíngyī), a skyscraper in Taipei, Taiwan.
  • Postcode of the "old town" in Reykjavík and title of the film 101 Reykjavík
  • Title of a concert film by DA Pennebaker about Depeche Mode
  • Number of Dalmatians in the book Hundertundein Dalmatiner or its Disney film adaptation 101 Dalmatians
  • In the USA the term for a basic course and, referring to it, the term for basic knowledge in a discipline.
• 108
  • Sacred number in Hinduism and Buddhism
• 110
  • Police emergency number in Germany
  • Number of cards for rummy and canasta
• 111
  • Sometimes used as a direct dialing in telephone exchange systems for telephone switching because it can be dialed quickly with a rotary dial
• 112
  • Euro (telephone) emergency call since 1991, with special functions on the GSM mobile phone
  • Emergency number for the fire brigade in Germany
• 114
  • Number of suras in the Koran as well as the logia of the Gospel of Thomas
• 115
  • Uniform authority telephone number in parts of Germany
• 117
  • Emergency number in Switzerland
• 118
  • Number of chemical elements currently (2015) detected
• 122
  • Emergency number for fire brigade in Austria (landline and GSM mobile networks)
• 128
  • Number of characters in a 7-bit code ( ASCII )
• 133
  • Emergency number for the police in Austria (landline and GSM mobile networks)
• 137.035 999 76 (50)
  • Reciprocal value of the fine structure constant
• 144
  • 1 Gros
  • 12 dozen
  • The height of the wall of the New Jerusalem is 144 cubits in Revelation. 21.17
  • Emergency number for rescue in Austria and Switzerland (landline and GSM mobile networks)
• 147
  • Maximum break , but not the highest possible break in snooker
• 150
  • Number of Psalms
• 153
  • Christian number symbolism: according to the Gospel of John ( Joh 21,11  EU ) number of fish caught by Peter as a symbol for all of humanity
  • According to the numerology of Pythagoras , the sum of all species in nature is 153
• 156
  • Product of a dozen (12) and a "dozen of the devil" (13)
• 165
  • Number of cards in Samba Canasta
• 168
  • Number of hours in a calendar week
• 170
  • Highest possible finish with darts in "Double-Out" mode
• 175
  • Until the early 1980s, it was a code word for homosexuals , alluding to Section 175, which at the time was still in the German Criminal Code - StGB - and which sometimes made homosexuality a criminal offense.
• 187
  • Stands for murder or death threat; comes from the US-American police, which codes under the abbreviation '187' murder cases
• 200
  • Numerical value of the Milesian- Greek number sigma
• 212
  • Boiling point of water in degrees Fahrenheit under normal conditions
• 235
  • Number of months in Meton's calendar cycle
• 256
  • Number of characters that can be represented with one byte
• 260
  • Number of days in a tonalamatl
• 300
  • Numerical value of the Milesian- Greek number tau
• 354
  • Number of days in a lunar year ( ) ${\displaystyle 6\times 29+6\times 30}$
• 360
  • Number of days in a year for the interest calculation (in Germany )
  • Numerical value of the full angle in degrees
  • Number of months in the Islamic calendar cycle
• 365
  • Number of days in the calendar year
• 365.24219 ... (Follow A155540 in OEIS )
  • Duration of the tropical year (which determines the seasons) in days
• 366
  • Number of days in the leap year
• 400
  • Numerical value of the full angle in gon
  • The civil Gregorian calendar repeats itself after 400 years (i.e. without the Easter date, but the same calendar date always falls on the same day of the week afterwards)
  • Numerical value of the Milesian- Greek number Ypsilon
• 420
  • 420, 4:20 or 4/20 (pronunciation four-twenty) is a code word for regular cannabis use and is widely used to identify with cannabis culture
• 440
  • Since 1939 the standard concert pitch valid in many countries has been fixed at 440 Hz
• 451
  • Alleged auto-ignition temperature of paper in degrees Fahrenheit , in the novel Fahrenheit 451 by Ray Bradbury
• 500
  • Greatest value of a euro banknote
  • Numerical value of the Milesian- Greek number Phi
  • Roman numeral D.
• 532
  • The Julian calendar repeats itself after 532 years
• 555
  • See 555 (telephone number)
• 600
  • Numerical value of the Milesian- Greek number Chi
• 613
  • Commandments in the Torah
• 616
  • In some old Bible manuscripts instead of the 666 in Revelation. 13.18
• 666
  • Biblically the "number of the beast" or Antichrist (Rev. 13:18)
  • Number of satanists and the devil
• 700
  • Numerical value of the Milesian- Greek number Psi
• 777
  • Mystically / biblically the "divine number"; with the importance of absolute perfection
• 800
  • Numerical value of the Milesian- Greek number omega
• 888
  • "Christ number", numerical value of the name Jesus (see also the meaning of the number 8 )
• 911
  • Emergency number in North America
  • 911 also stands for the terrorist act of September 11, 2001 (9/11)
• 940
  • Number of months in a Callipean cycle
• 969
  • Age of Methuselah , the oldest man mentioned in the Bible in Genesis 1 ( Genesis , 5: 21-27)
• 1000
  • In the Bible in chapter 20 of Revelation, Millennial Kingdom of Christ; also in National Socialist rhetoric
  • Roman numeral M

Up to 10,000

• 1,001
  • Arabic magic number (for example "Tales from the Arabian Nights ")
• 1,024
  • Basis for the IEC binary prefixes . 1 KiB = byte = byte ${\displaystyle 2^{10}}$ ${\displaystyle 1024^{1}}$
• 1,080
  • Number of chalakim , the time units of an hour in the Jewish calendar (about 3.33 s)
• 1,154
  • Number of complete tiling of a regular decagon with the Penrose diamonds (36 °; 144 ° and 72 °; 108 °) and the Mukundi crown (concave pentagon (36 °; 108 °; 252 °; 108 °; 36 °)) , whereby two tiling are considered to be different if and only if they cannot be converted into one another by rotation
• 1,189
  • Number of chapters in the Bible
• 1,337
  • Frequently used abbreviation for Leetspeak
  • Jokingly in "modern number mysticism" also 42 ${\displaystyle (\pi \times 1337)/100=}$
• 1,435
  • Standard gauge of the railway in millimeters
• 1,440
  • Number of minutes in a day
  • Number of kilobytes on a normally formatted 3.5 ″ floppy disk
• 2,701
  • Important number in the cryptonomicon
• 6,666
  • Alleged number of Āyāt in the Quran
• 6,585.32
  • Length of a Saros cycle in days - see eclipse cycles
• 7,200
  • Number of days in a katun period in the Maya calendar
• 8,766
  • Number of hours in a year according to the Julian calendar
• 10,000
  • A myriad
  • Ten thousand years / Banzai

Up to 1 million

• 10,631
  • Number of days in an Islamic period
• 12,000
  • biblical : length, width and height of the New Jerusalem in Revelation. 21.16 are 12,000 stadia
• 18,980
  • Is - the number of days in the Mayan calendar period ${\displaystyle 52\times 365}$
• 27,759
  • Number of days in the Callipean cycle
• 31,169
  • Number of verses in the Bible
• 44,760
  • Number of Warriors by Reuben (1 Chr 5:18)
• 86,400
  • Number of seconds in a day
• 144,000
  • Mystical / Biblical number of the saved on Judgment Day ; derived from " humans" or 12,000 sons from each of the 12 tribes of Israel (Rev 7,4) ${\displaystyle 12\times 12\times 1000}$
• 146.097
  • Number of days in the 400-year Gregorian calendar cycle
• 304.805
  • Number of letters in the Torah
• 525,600
  • Number of minutes in a year
• 604,800
  • Number of seconds in a week

Up to 1 billion

• 1,048,576
  • 1 MiB = byte = byte ${\displaystyle 2^{20}}$ ${\displaystyle 1024^{2}}$
• 3,674,160
  • Number of positions of a Rubik's cube of the size (pocket cube) that can be reached by manual rotation ${\displaystyle 2\times 2\times 2}$
• 3,447,360
  • Number of years in the Jewish calendar cycle
• 5,700,000
  • Number of years in the Gregorian Easter cycle (after that, Easter is always on the same date)
• 8,145,060
  • Number of possibilities in the Swiss and Austrian number lottery "6 out of 45"; the probability of a "six" is 1 in 8,145,060
• 10,518,300
  • Number of possible combinations for a player's card hand at Schafkopf
• 13,983,816
  • Number of possible combinations in the German lottery "6 out of 49"
• 16,777,216
  • ${\displaystyle 2^{24}}$; Use in IT , e.g. B. the number of possible color gradations with 24 bit color depth
• 76.275.360
  • Number of possibilities in the Euro-Millions Lotto: 5 out of 50 numbers and 2 out of 9 stars
• 299,792,458
  • The speed of light in a vacuum , defined in m / s

Over 1 billion

• 1,073,741,824
  • 1 GiB = byte = byte ${\displaystyle 2^{30}}$ ${\displaystyle 1024^{3}}$
• 3.101.788.170
  • Number of DNA base pairs that encode human genetic information (≙ 740 MiB ), approximate value
• 3,735,928,559
  • The numerical value results in the character string DEADBEEF in the hexadecimal system.
• 4,294,967,296
  • Number of possible IP addresses according to the IPv4 protocol: ${\displaystyle (2^{8})^{4}=2^{32}}$
• 149,597,870,691
  • Length of the astronomical unit (AU) in meters; Mean distance from earth to sun in meters
• 1,099,511,627,776
  • 1 TiB = byte = byte ${\displaystyle 2^{40}}$ ${\displaystyle 1024^{4}}$
• 2,753,294,408,504,640
  • Number of all possible card distributions in the game of Skat
• 9,460,730,472,580,800
  • One light year (in meters)
• 99,561,092,450,391,000
  • Number of possible card distributions at Schafkopf
• 710,609,175,188,282,000 to 1
  • The likelihood that Mikhail Gorbachev is the Antichrist . Calculated by Robert W. Faid and the Ig Nobel Prize 1993 awarded
• 18,446,744,073,709,551,615
  • ( ) ${\displaystyle {2^{64}}-1}$
  • Number of grains of wheat that Sissa ibn Dahir was supposed to receive from the Indian ruler Shihram for the invention of the game of chess , according to the wheat grain legend
• 43,252,003,274,489,856,000
  • Number of positions of a Rubik's cube of the size that can be achieved by manual rotation ${\displaystyle 3\times 3\times 3}$
• 2,248,575,441,654,260,591,964
  • Number of all possible card distributions with a double head with nines.
• 6,670,903,752,021,072,936,960
  • Number of possible Sudoku puzzles ( ) ${\displaystyle 9\times 9}$
• 6.022 140 76 · 10 ²³
  • Avogadro constant , number of particles (atoms or molecules) per amount of substance ( mol )
• 60.176.864.903.260.346.841.600.000
  • Number of possible starting positions (" key space ") of the Enigma-M4 , the cryptographically strongest Enigma cipher machine used in World War II
• 340,282,366,920,938,463,463,374,607,431,768,211,456
  • Number of possible IP addresses according to the IPv6 protocol: ${\displaystyle (2^{16})^{8}=2^{128}}$
• 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000
  • ( ) ${\displaystyle \approx 7{,}401\cdot 10^{45}}$
  • Number of positions of a Rubik's cube of the size ( Master Cube ) that can be reached by manual rotation ${\displaystyle 4\times 4\times 4}$
• 81,171,437,193,104,932,746,936,103,027,318,645,818,654,720,000
  • ( ) ${\displaystyle \approx 8{,}11714\cdot 10^{46}}$
  • Number of possible Sudoku puzzles ( ) ${\displaystyle 12\times 12}$
• 282,870,942,277,741,856,536,180,333,107,150,328,293,127,731,985,672,134,721,536,000,000,000,000,000
  • ( ) ${\displaystyle \approx 2{,}82871\cdot 10^{74}}$
  • Number of positions of a Rubik's cube of the size ( Professor's Cube ) that can be reached by manual rotation ${\displaystyle 5\times 5\times 5}$
• 10 ¹⁰⁰
  • A googol
• 19,500,551,183,731,307,835,329,126,754,019,748,794,904,992,692,043,434,567,152,132,912,323,232,706,135,469,180.065,278,712,755,853,360,682,328,551,719.137,311,299,993,600,000,000,000,000,000,000,000. 000,000,000,000
  • ( ) ${\displaystyle \approx 1{,}95006\cdot 10^{160}}$
  • Number of positions of a Rubik's cube of the size ( V-Cube 7 ) that can be reached by manual rotation ${\displaystyle 7\times 7\times 7}$
• 10 ⁶⁰⁰⁰ -1
  • (a number out of 6000 nines): the highest number that can be named with a classic number name (according to the long scale ). The next number ( , a 1 with 6000 zeros) should (again) be called “Millinillion”. The correct classic name would be many pages long. ${\displaystyle 10^{6000}}$ ${\displaystyle 10^{6000}-1}$
• 10 ^googol = ${\displaystyle 10^{(10^{100})}}$
  • A googolplex
• 10 ^Googolplex
  • A googolplexplex, also called googolplexian
• 10 ^{Googolplexplex}
  • ${\displaystyle 10^{10^{10^{(10^{100})}}}}$ Googolplexplexplex
• 10 ^{Googolplexplexplex}
  • ${\displaystyle 10^{10^{10^{10^{(10^{100})}}}}}$ Googolplexplexplexplex

literature

• Walter Kranzer: Mathematics is that interesting . Aulis Verlag , Cologne 1989, ISBN 3-7614-0856-0 .
• F. Le Lionnais: Les Nombres Remarquables . Hermann, Paris 1983
• David Wells: The Lexicon of Numbers . Fischer, Frankfurt am Main 1991, ISBN 3-596-10135-2

Historical literature

• Wilhelm Heinrich Roscher : The number 50 in myth, cult, epic and tactics of the Hellenes and other peoples, especially the Semites. Leipzig 1917 (= treatises of the Saxon Society of Sciences: philological-historical class , 33, 5).

See also

• Number names
• Mathematical constant
• Numerology
• Scales of the order of magnitude of various elementary sizes
• Interesting Numbers Paradox

Web links

• What's Special About This Number?
• The secret lives of numbers

Individual evidence

1. ↑ Follow A004023 in OEIS
2. ↑ Cohn, Jhon E., Square Fibonacci Numbers, etc., Bedford Col lege, University of London, London, NWI http://www.fq.math.ca/Scanned/2-2/cohn2.pdf
3. ↑ Follow A046253 in OEIS
4. ↑ spiegel.de
5. ↑ nature.com
6. ↑ State mathematics competition 2005/2006 Bavaria (accessed on June 19, 2010)
7. ↑ Weisstein, Eric W .: Ramanujan Constant. Wolfram MathWorld, accessed December 4, 2015 . 
8. ↑ Kranzer: p. 144.
9. ↑ Eric W. Weisstein : Skewes Number . In: MathWorld (English).
10. ↑ Under favorable visibility conditions, Uranus and the minor planet (4) Vesta (asteroid) are also visible to the naked eye.
11. ↑ From its discovery in 1930 to the redefinition of the term planet in 2006, Pluto was considered the ninth planet in our solar system.
12. ↑ actually there are fewer, see kuranyolunda.com
13. ↑ Mathematical semester reports Volume 56, Number 2, 177–185, doi: 10.1007 / s00591-009-0056-8 Mathematics in research and application Card distributions for Skat, Doppelkopf, Rummy and Canasta Jens-P. Bode and Arnfried Kemnitz