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The Pippenger product is an unexpected Wallis-like formula for given by(1)(OEIS A084148 and A084149; Pippenger 1980). Here, the th term for is given by(2)(3)where is a double factorial and is the gamma function.

Compass and straightedge geometric constructions dating back to Euclid were capable of inscribing regular polygons of 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, ..., sides. In 1796 (when he was 19 years old), Gauss gave a sufficient condition for a regular -gon to be constructible, which he also conjectured (but did not prove) to be necessary, thus showing that regular -gons were constructible for , 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, ... (OEIS A003401).A complete enumeration of "constructible" polygons is given by those with central angles corresponding to so-called trigonometry angles.Gardner (1977) and independently Watkins (Conway and Guy 1996, Krížek et al. 2001) noticed that the number of sides for constructible polygons with odd numbers of sides are given by the first 32 rows of the Sierpiński sieve interpreted as binary numbers, giving 1, 3, 5, 15, 17, 51, 85,..

Consider the Euler product(1)where is the Riemann zeta function and is the th prime. , but taking the finite product up to , premultiplying by a factor , and letting gives(2)(3)where is the Euler-Mascheroni constant (Havil 2003, p. 173). This amazing result is known as the Mertens theorem.At least for , the sequence of finite products approaches strictly from above (Rosser and Schoenfeld 1962). However, it is highly likely that the finite product is less than its limiting value for infinitely many values of , which is usually the case for any such inequality due to the presence of zeros of on the critical line . An example is Littlewood's famous proof that the sense of the inequality , where is the prime counting function and is the logarithmic integral, reverses infinitely often. While Rosser and Schoenfeld (1962) suggest that "perhaps one can extend [this] result to show that [the Mertens inequality] fails for large ; we have not investigated..

The th Beraha constant (or number) is given by is , where is the golden ratio, is the silver constant, and . The following table summarizes the first few Beraha numbers.approx.1420314252.6186373.24783.41493.532103.618Noninteger Beraha numbers can never be roots of any chromatic polynomials with the possible exception of (G. Royle, pers. comm., Nov. 21, 2005). However, the roots of chromatic polynomials of planar triangulations appear to cluster around the Beraha numbers (and, technically, are conjectured to be accumulation points of roots of planar triangulation chromatic polynomials).

Consider an (0, 1)-matrix such as(1)for . Call two elements adjacent if they lie in positions and , and , or and for some . Call the number of such arrays with no pairs of adjacent 1s. Equivalently, is the number of configurations of nonattacking kings on an chessboard with regular hexagonal cells.The first few values of for , 2, ... are 2, 6, 43, 557, 14432, ... (OEIS A066863).The hard square hexagon constant is then given by(2)(3)(OEIS A085851).Amazingly, is algebraic and is given by(4)where(5)(6)(7)(8)(9)(10)(11)(Baxter 1980, Joyce 1988ab).The variable can be expressed in terms of the tribonacci constant(12)where is a polynomial root, as(13)(14)(15)(T. Piezas III, pers. comm., Feb. 11, 2006).Explicitly, is the unique positive root(16)where denotes the th root of the polynomial in the ordering of the Wolfram Language...

The limiting rabbit sequence written as a binary fraction (OEIS A005614), where denotes a binary number (a number in base-2). The decimal value is(1)(OEIS A014565).Amazingly, the rabbit constant is also given by the continued fraction [0; , , , , ...] = [2, 2, 4, 8, 32, 256, 8192, 2097152, 17179869184, ...] (OEIS A000301), where are Fibonacci numbers with taken as 0 (Gardner 1989, Schroeder 1991). Another amazing connection was discovered by S. Plouffe. Define the Beatty sequence by(2)where is the floor function and is the golden ratio. The first few terms are 1, 3, 4, 6, 8, 9, 11, ... (OEIS A000201). Then(3)This is a special case of the Devil's staircase function with .The irrationality measure of is (D. Terr, pers. comm., May 21, 2004).

Let denote the partition lattice of the set . The maximum element of is(1)and the minimum element is(2)Let denote the number of chains of any length in containing both and . Then satisfies the recurrence relation(3)where and is a Stirling number of the second kind. The first few values of for , 2, ... are then 1, 1, 4, 32, 436, 9012, 262760, ... (OEIS A005121).Lengyel (1984) proved that the quotient(4)is bounded between two constants as , and Flajolet and Salvy (1990) improved the result of Babai and Lengyel (1992) to show that(5)(OEIS A086053).

Plouffe's constants are numbers arising in summations of series related to where is a trigonometric function. Define the Iverson bracket function(1)Now define through(2)(3)then(4)(5)(6)(OEIS A086201).For(7)(8)the sum is (amazingly) given by(9)(10)(11)(OEIS A086202), where denotes the XOR of binary digits (Chowdhury 2001a; Finch 2003, p. 432). A related sum is given by(12)(13)(14)(OEIS A111953), where again denotes the XOR of binary digits (Chowdhury 2001b; Finch 2005, p. 20).Letting(15)(16)then(17)(18)(OEIS A049541).Plouffe asked if the above processes could be "inverted." He considered(19)(20)giving(21)(22)and(23)(24)giving(25)(26)and(27)(28)giving(29)(30)(31)(OEIS A086203), where the identity was conjecturedby Plouffe and proved by Borwein and Girgensohn (1995). is sometimes known as Plouffe's constant (Plouffe 1997), although this angle had arisen in the geometry of the..

Gieseking's constant is defined by(1)(2)(3)(4)(5)(6)(7)(OEIS A143298), where is Clausen's integral, is a dilogarithm, and is a trigamma function.

The Pell constant is the infinite product(1)(2)(3)(OEIS A141848), where is a q-Pochhammer symbol.The Pell constant is irrational, but it is notknown if it is transcendental (Stevenhagen 1993; Finch 2003, p. 119).

The paper folding constant is the constant given by(1)(2)(3)(OEIS A143347).

The Weierstrass constant is defined as the value , where is the Weierstrass sigma function with half-periods and . Amazingly, it has the closed form (1)(2)(OEIS A094692), where is the gamma function.

A constant appearing in formulas for the efficiency ofthe Euclidean algorithm,(1)(2)(OEIS A143304), where is Porter's constant.

Consider the problem of comparing two real numbers and based on their continued fraction representations. Then the mean number of iterations needed to determine if or is given by(1)(2)(3)(4)(OEIS A143303; Finch 2003, p. 161), where is a polylogarithm.Let , then the first few values can be given by(5)(6)(7)(8)(9)(10)(11)(12)The value(13)(OEIS A143302) is then known as the Valléeconstant (Finch 2003, p. 161).

Let be an arbitrary trigonometric polynomial(1)with real coefficients, let be a function that is integrable over the interval , and let the th derivative of be bounded in . Then there exists a polynomial for which(2)for all , where is the smallest constant possible, known as the th Favard constant. can be given explicitly by the sum(3)which can be written in terms of the Lerch transcendentas(4)These can be expressed by(5)where is the Dirichlet lambda function and is the Dirichlet beta function. Explicitly,(6)(7)(8)(9)(10)(11)(OEIS A050970 and A050971).

Consider the sequence of partial sums defined by(1)As can be seen in the plot above, the sequence has two limit points at and 0.187859... (which are separated by exactly 1). The upper limit point is sometimes known as the MRB constant after the initials of its original investigator (Burns 1999; Plouffe).Sums for the MRB constant are given by(2)(3)(4)(5)(6)(Finch 2003, p. 450; OEIS A037077).The constant can also be given as a sum over derivatives of the Dirichlet eta function as(7)(8)where(9)and denotes the th derivative of evaluated at (Crandall 2012ab).An integral expression for the constant is given by(10)(M. Burns, pers. comm., Jan. 21, 2020).No closed-form expression is known for this constant (Finch 2003, p. 450).

The "ternary" Champernowne constant can be defined by concatenating the ternary representations of the integers(1)(2)(OEIS A054635 and A077771). This has continued fraction [0, 1, 1, 2, 37, 1, 162, 1, 1, 1, 3, 1, 7, 1, 9, 2, 3, 1, 3068518062211324, 2, 1, ...] (OEIS A077772), which like the normal Champernowne constant, displays sporadic large terms.

Let(1)(OEIS A064853) be the arc length of a lemniscate with . Then the lemniscate constant is the quantity(2)(3)(4)(5)(6)(OEIS A062539; Abramowitz and Stegun 1972; Finch 2003, p. 420), where is a complete elliptic integral of the first kind. Todd (1975) cites T. Schneider as proving to be a transcendental number in 1937.The quantity(7)(8)(OEIS A085565; Le Lionnais 1983) is sometimesknown as the first lemniscate constant, while(9)(10)(OEIS A076390), where is the reciprocal of Gauss's constant, is sometimes known as the second lemniscate constant (Todd 1975, Gosper 1976, Lewanowicz and Paszowski 1995).

The constantswhere and are th and th order polynomials, and is the set of all rational functions with real coefficients.

"The" Smarandache constant is the smallest solution to the generalized Andrica's conjecture, (OEIS A038458).The first Smarandache constant is defined as(1)(OEIS A048799), where is the Smarandache function. Cojocaru and Cojocaru (1996a) prove that exists and is bounded by .Cojocaru and Cojocaru (1996b) prove that the second Smarandache constant(2)(OEIS A048834) is an irrationalnumber.Cojocaru and Cojocaru (1996c) prove that the series(3)converges to a number , and that(4)converges for a fixed real number . The values for small are(5)(6)(7)(8)(9)(OEIS A048836, A048837,and A048838).Sandor (1997) shows that the series(10)converges to an irrational. Burton (1995) andDumitrescu and Seleacu (1996) show that the series(11)converges. Dumitrescu and Seleacu (1996) show that the series(12)and(13)converge for a natural number (which must be nonzero in the latter case). Dumitrescu and Seleacu (1996) show that(14)converges...

Let the sum of squares function denote the number of representations of by squares, then the summatory function of has the asymptotic expansion(1)where(2)(3)(4)(5)(OEIS A241017) is the Sierpiński constant (Finch 2003, p. 123), is the Dirichlet beta function, is the Euler-Mascheroni constant, and is the gamma function.

Consider the sum(1)where the s are nonnegative and the denominators are positive. Shapiro (1954) asked if(2)for all . It turns out (Mitrinovic et al. 1993) that this inequality is true for all even and odd .Define(3)and let(4)Then Rankin (1958) proved that(5) can be computed by letting be the function convex hull of the functions(6)(7)Then(8)(OEIS A086277; Drinfeljd 1971).A modified sum was considered by Elbert (1973):(9)Consider(10)where(11)and let be the convex hull of(12)(13)Then(14)(OEIS A086278).

Define with positive as(1)Then(2)as increases, where the Shallit constant is(3)(OEIS A086276; Shallit 1995). In their solution,Grosjean and De Meyer (quoted in Shallit 1995) reduced the complexity of the problem.

Cahen's constant is defined as(1)(2)(OEIS A118227), where is the th term of Sylvester's sequence.

The sum of reciprocal multifactorials can be givenin closed form by the beautiful formula(1)(2)(3)where is a lower incomplete gamma function (E. W. Weisstein, Aug. 6, 2008).This gives the special cases(4)(5)

Let be the set of complex analytic functions defined on an open region containing the set closure of the unit disk satisfying and . For each in , let be the supremum of all numbers such that there is a subregion in on which is one-to-one and such that contains a disk of radius . In 1925, Bloch (Conway 1989) showed that .Define Bloch's constant by(1)Ahlfors and Grunsky (1937) derived(2)Bonk (1990) proved that , which was subsequently improved to (Chen and Gauthier 1996; Xiong 1998; Finch 2003, p. 456).Ahlfors and Grunsky (1937) also conjectured that the upper limit is actually the value of ,(3)(4)(5)(OEIS A085508; Le Lionnais 1983).

The Gelfond-Schneider constant is sometimesknown as the Hilbert number.Flannery and Flannery (2000, p. 35) define a Hilbert number as a positive integer of the form (i.e., a positive integer such that ). The first few are then 1, 5, 9, 13, 17, 21, 25, ... (OEIS A016813). A Hilbert number that is not divisible by a smaller Hilbert number (other than 1) is then called a Hilbert prime (or S-prime; Apostol 1976, p. 101); otherwise, is called a Hilbert composite. The first few Hilbert primes are 5, 9, 13, 17, 21, 29, 33, 37, 41, 49, ... (OEIS A057948), and the first few Hilbert composites are 25, 45, 65, 81, 85, ... (OEIS A054520).Factorization with respect to Hilbert primes is not necessarily unique, as illustrated by the exampleThe first few such examples are 441, 693, 1089, 1197, 1449, ... (OEIS A057949)...

The "binary" Champernowne constant is obtained by concatenating the binary representations of the integers(1)(2)(OEIS A030190 and A066716). The sequence given by the first few concatenations is therefore 1, 110, 11011, 11011100, 11011100101, ... (OEIS A058935). can also be written(3)with(4)and the floor function (Bailey and Crandall 2002). Interestingly, is 2-normal (Bailey and Crandall 2002). has continued fraction [0, 1, 6, 3, 1, 6, 5, 3, 3, 1, 6, 4, 1, 3, 298, 1, 6, 1, 1, 3, 285, 7, 2, 4, 1, 2, 1, 2, 1, 1, 4534532, ...] (OEIS A066717), which exhibits sporadic large terms. The numbers of decimal digits in these terms are 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, ....

Let be the error of the best uniform approximation to a real function on the interval by real polynomials of degree at most . If(1)then Bernstein showed that(2)He conjectured that the lower limit () was . However, this was disproven by Varga and Carpenter (1987) and Varga (1990), who computed(3)For rational approximations for and of degree and , D. J. Newman (1964) proved(4)for . Gonchar (1967) and Bulanov (1975) improved the lower bound to(5)Vjacheslavo (1975) proved the existence of positive constants and such that(6)(Petrushev 1987, pp. 105-106). Varga et al. (1993) conjectured and Stahl(1993) proved that(7)

(1)(2)(3)(OEIS A073003), where is the exponential integral. Stieltjes showed it has the continued fraction representation(4)

Van der Corput's constant is given by(1)(2)(3)(OEIS A143305), where and are Fresnel integrals,(4)and is the transcendental root of(5)(6)with , namely(7)(OEIS A143306).

Quinn et al. (2007) investigated a class of coupled oscillators whose bifurcation phase offset had a conjectured asymptotic behavior of , with an experimental estimate for the constant as (OEIS A131329). Rather amazingly, Bailey et al. (2007) were able to find a closed form for as the unique root of in the interval , where is a Hurwitz zeta function.A related constant conjectured by Quinn et al. (2007) to exist was definedin terms of(1)and given by(2)(OEIS A131330). Even more amazingly, the exact value of this constant was also found by Bailey et al. (2007) without full proof, but with enough to indicate that such a proof could in principle be constructed, to have the exact value(3)

A Trott constant is a real number whose decimal digits are equal to the terms ofits continued fraction.The first Trott constant (OEIS A039662) was discovered by M. Trott in 1999. While it is theoretically possible to extend this sequence arbitrarily far, it is impractical to do so since agreement after 639 terms is so close that the number of consecutive term pairs of "90" that would immediately follow the 639th term would exceed (Schoenfield 2010).The second Trott constant is the number (OEIS A091694; Trott 2004, p. 70) which is equal to its non-simple continued fractionThe third Trott constant is the number (OEIS A113307; M. Trott, pers. comm., Oct. 24, 2005) which is equal to its non-simple continued fractionVery little seems to be known about the existence and uniqueness of such numbers...

For a real number , let be the number of terms in the convergent to a regular continued fraction that are required to represent decimal places of . Then Lochs' theorem states that for almost all ,(1)(2)(3)(OEIS A086819; Lochs 1964). This number issometimes known as Lochs' constant.The reciprocal of this constant is(4)(5)(OEIS A062542; Finch 2003, p. 60).Lochs' constant is related to the Lévy constant by(6)(7)In the index and table of constants Finch (2003, pp. 546 and 596) refers to the quantity(8)related to Porter's constant as "Lochs'constant," though this terminology appears to be nonstandard.

Let be the simple continued fraction of a "generic" real number, where the numbers are the partial quotients. Then the Khinchin (or Khintchine) harmonic mean(1)defined analogously to the Khinchin constant but with the partial quotients taken to the power, exists and has a unique common value (except for a set of real numbers with measure zero) given by(2)(3)(4)(OEIS A087491; Bailey et al. 1997, Plouffe).Khinchin's constant and the Khinchin harmonic mean are just two of an infinite family of such constants , the first few of which are summarized in the following table.OEISvalue0A0022102.685452001065306445309714835481795693820382293994462A0874911.745405662407346863494596309683661067294936618777984A0874921.450340328495630406052983076680697881408299979605904A0874931.313507078687985766717339447072786828158129861484792A0874941.236961809423730052626227244453422567420241131548937A0874951.189003926465513154062363732771403397386092512639671A0874961.156552374421514423152605998743410046840213070718761A0874971.133323363950865794910289694908868363599098282411797A0874981.115964408978716690619156419345349695769491182230400A0874991.102543136670728013836093402522568351022221284149318A0875001.091877041209612678276110979477638256493272651429656..

The golden gnomon is the obtuse isosceles triangle whose ratio of side to base lengths is given by , where is the golden ratio. Such a triangle has angles of -- and can be constructed from a regular pentagon as illustrated above in red. The corresponding 36-72-72 triangle with side-to-base ratio is a golden triangle.Golden triangles and gnomons can be dissected into smaller triangles that are golden gnomons and golden triangles (Livio 2003, p. 79).

A golden rhombohedron is a rhombohedron whose faces consist of congruent golden rhombi. Golden rhombohedra are therefore special cases of a trigonal trapezohedron as well as zonohedra.There are two distinct golden rhombohedra: the acute golden rhombohedron and obtuse golden rhombohedron. Both are built from six golden rhombi and comprise two of the five golden isozonohedra. These polyhedra are implemented in the Wolfram Language as PolyhedronData["AcuteGoldenRhombohedron"] and PolyhedronData["ObtuseGoldenRhombohedron"], respectively.The acute and obtuse golden rhombohedra with edge length both have surface area(1)and have volumes(2)(3)respectively.

For every positive integer , there is a unique finite sequence of distinct nonconsecutive (not necessarily positive) integers , ..., such that(1)where is the golden ratio.For example, for the first few positive integers,(2)(3)(4)(5)(6)(7)(8)(OEIS A104605).The numbers of terms needed to represent for , 2, ... are given by 1, 2, 2, 3, 3, 3, 2, 3, 4, 4, 5, 4, ... (OEIS A055778), which are also the numbers of 1s in the base- representation of .The following tables summarizes the values of that require exactly powers of in their representations.OEISnumbers requiring exactly powers2A0052482, 3, 7, 18, 47, 123, 322, 843, ...3A1046264, 5, 6, 8, 19, 48, 124, 323, 844, ...4A1046279, 10, 12, 13, 14, 16, 17, 20, 21, 25, ...5A10462811, 15, 22, 23, 24, 26, 30, 31, 32, 34, ...

The Fibonacci factorial constant is the constant appearing in the asymptotic growth of the fibonorials (aka. Fibonacci factorials) . It is given by the infinite product(1)where(2)and is the golden ratio.It can be given in closed form by(3)(4)(5)(OEIS A062073), where is a q-Pochhammer symbol and is a Jacobi theta function.

The golden triangle, sometimes also called the sublime triangle, is an isosceles triangle such that the ratio of the hypotenuse to base is equal to the golden ratio, . From the above figure, this means that the triangle has vertex angle equal to(1)or , and that the height is related to the base through(2)(3)(4)The inradius of a golden triangle is(5)The triangles at the tips of a pentagram (left figure) and obtained by dividing a decagon by connecting opposite vertices (right figure) are golden triangles. This follows from the fact that(6)for a pentagram and that the circumradius of a decagon of side length is(7)Golden triangles and gnomons can be dissected into smaller triangles that are golden gnomons and golden triangles (Livio 2002, p. 79).Successive points dividing a golden triangle into golden gnomons and triangles lieon a logarithmic spiral (Livio 2002, p. 119).Kimberling (1991) defines a second type of golden triangle..

The golden ratio conjugate, also called the silver ratio, is the quantity(1)(2)(3)(4)(5)(OEIS A094214), where is the golden ratio.

Somos's quadratic recurrence constant is defined via the sequence(1)with . This has closed-form solution(2)where is a polylogarithm, is a Lerch transcendent. The first few terms are 1, 2, 12, 576, 1658880, 16511297126400, ... (OEIS A052129). The terms of this sequence have asymptotic growth as(3)(OEIS A116603; Finch 2003, p. 446, term corrected), where is known as Somos's quadratic recurrence constant. Here, the generating function in satisfies the functional equation(4)Expressions for include(5)(6)(7)(8)(9)(OEIS A112302; Ramanujan 2000, p. 348;Finch 2003, p. 446; Guillera and Sondow 2005).Expressions for include(10)(11)(12)(13)(14)(OEIS A114124; Finch 2003, p. 446; Guillera and Sondow 2005; J. Borwein, pers. comm., Feb. 6, 2005), where is a polylogarithm. is also given by the unit square integral(15)(16)(Guillera and Sondow 2005).Ramanujan (1911; 2000, p. 323) proposed..

The golden ratio can be written in terms of a nested radical in the beautiful form(1)which can be written recursively as(2)for , with .Paris (1987) proved approaches at a constant rate, namely(3)as , where(4)(OEIS A105415) is the Paris constant.A product formula for is given by(5)(Finch 2003, p. 8).Another formula is given by letting be the analytic solution to the functional equation(6)for , subject to initial conditions and . Then(7)(Finch 2003, p. 8).A close approximation is , which is good to 4 decimal places (M. Stark, pers. comm.).

A problem listed in a fall issue of Gazeta Matematică in the mid-1970s posed the question if and(1)for , 2, ..., then are there any values for which ? The problem, listed as one given on an entrance exam to prospective freshman in the mathematics department at the University of Bucharest, was solved by C. Foias.It turns out that there exists exactly one real number(2)(OEIS A085848) such that if , then . However, no analytic form is known for this constant, either as the root of a function or as a combination of other constants. Moreover, in this case,(3)which can be rewritten as(4)where is the prime counting function. However, Ewing and Foias (2000) believe that this connection with the prime number theorem is fortuitous.Foias also discovered that the problem stated in the journal was a misprint of the actual exam problem, which used the recurrence (Ewing and Foias 2000). In this form, the recurrence converges to(5)(OEIS A085846), which..

The twin primes constant (sometimes also denoted ) is defined by(1)(2)(3)(4)where the s in sums and products are taken over primes only. This can be written as(5)where is the prime zeta function.Flajolet and Vardi (1996) give series with accelerated convergence(6)(7)with(8)where is the Möbius function. The values of for , 2, ... are 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, ... (OEIS A001037). Equation (7) has convergence like . was computed to 45 digits by Wrench (1961) and Gourdon and Sebah list 60 digits.(9)(OEIS A005597). Le Lionnais (1983, p. 30) calls the Shah-Wilson constant, and the twin prime constant (Le Lionnais 1983, p. 37).

The characteristic function(1)of primes has values 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, ... (OEIS A010051) for , 2, .... The constant obtained by concatenating these digits and interpreting them as a binary fraction is therefore(2)(3)(OEIS A051006).The continued fraction is [0, 2, 2, 2, 3, 12, 131, 1, ...] (OEIS A051007). It has high-water marks of 0, 2, 3, 12, 131, 169, ... (OEIS A102878), occurring at positions 0, 1, 4, 5, 6, 20, 31, 54, ... (OEIS A103313).

Given a positive integer , let its prime factorization be written(1)Define the functions and by , , and(2)(3)The first few terms of are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, ... (OEIS A051904), while the first few terms of are 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, ... (OEIS A051903).Then the average value of tends to(4)Here, the running average values are given by 1/2, 2/3, 3/4, 1, 1, 1, 1, 11/9, 13/10, 14/11, 5/4, 16/13, ... (OEIS A086195 and A086196).In addition, the ratio(5)where is the Riemann zeta function (Niven 1969).Niven (1969) also proved that(6)where Niven's constant is given by(7)(OEIS A033150). Here, the running average values are given by 1/2, 2/3, 3/4, 1, 1, 1, 1, 11/9, 13/10, 14/11, 5/4, 17/13, ... (OEIS A086197 and A086198).The continued fraction of Niven's constant is 1, 1, 2, 2, 1, 1, 4, 1, 1, 3, 4, 4, 8, 4, 1, ... (OEIS A033151). The positions at which the digits 1, 2, ... first occur in the continued fraction are 1, 3, 10, 7, 47, 41, 34,..

Murata's constant is defined as(1)(2)(OEIS A065485), where the product is over the primes . It can also be written as the sum(3)(4)where is the prime zeta function.

Taniguchi's constant is defined as(1)(2)(OEIS A175639), where the product is over the primes . Taking the logarithm, expand the sum about infinity, and then summing the terms gives a "closed" form as(3)(4)where is the prime zeta function and the s are rational numbers given as the coefficients of in the series(5)

Mills (1947) proved the existence of a real constant such that(1)is prime for all integers , where is the floor function. Mills (1947) did not, however, determine , or even a range for .A generalization of Mills' theorem to an arbitrary sequence of positiveintegers is given as an exercise by Ellison and Ellison (1985).The least such that is prime for all integers is known as Mills' constant.Mills' proof was based on the following theorem by Hoheisel (1930) and Ingham (1937). Let be the th prime, then there exists a constant such that(2)for all . This has more recently been strengthened to(3)(Mozzochi 1986). If the Riemann hypothesisis true, then Cramér (1937) showed that(4)(Finch 2003).Hardy and Wright (1979) and Ribenboim (1996) point out that, despite the beauty of such prime formulas, they do not have any practical consequences. In fact, unless the exact value of is known, the primes themselves must be known in advance to determine..

Mills' theorem states that there exists a real constant such that is prime for all positive integers (Mills 1947). While for each value of , there are uncountably many possible values of such that is prime for all positive integers (Caldwell and Cheng 2005), it is possible to define Mills' constant as the least such thatis prime for all positive integers , giving a value of(OEIS A051021). is therefore given by the next prime after , and the values of are known as Mills' primes (Caldwell and Cheng 2005).Caldwell and Cheng (2005) computed more than 6850 digits of assuming the truth of the Riemann hypothesis. Proof of primality of the 13 Mills prime in Jul. 2013 means that approximately digits are now known.It is not known if is irrational.

Let and be nonzero integers such that (except when ). Also let be the set of primes for which for some nonnegative integer . Then assuming the generalized Riemann hypothesis, Stephens (1976) showed that the density of relative to the primes is a rational multiple of(OEIS A065478), where is the th prime (Finch 2003).

The Skewes number (or first Skewes number) is the number above which must fail (assuming that the Riemann hypothesis is true), where is the prime counting function and is the logarithmic integral.Isaac Asimov featured the Skewes number in his science fact article "Skewered!"(1974).In 1912, Littlewood proved that exists (Hardy 1999, p. 17), and the upper boundwas subsequently found by Skewes (1933). The Skewes number has since been reduced to by Lehman in 1966 (Conway and Guy 1996; Derbyshire 2004, p. 237), by te Riele (1987), and less than (Bays and Hudson 2000; Granville 2002; Borwein and Bailey 2003, p. 65; Havil 2003, p. 200; Derbyshire 2004, p. 237). The results of Bays and Hudson left open the possibility that the inequality could fail around , and thus established a large range of violation around (Derbyshire 2004, p. 237). More recent work by Demichel establishes that the first crossover..

(1)(2)(OEIS A093827), where is the totient function and is the divisor function.

The Mertens constant , also known as the Hadamard-de la Vallee-Poussin constant, prime reciprocal constant (Bach and Shallit 1996, p. 234), or Kronecker's constant (Schroeder 1997), is a constant related to the twin primes constant and that appears in Mertens' second theorem,(1)where the sum is over primes and is a Landau symbol. This sum is the analog of(2)where is the Euler-Mascheroni constant (Gourdon and Sebah).The constant is given by the infinite sum(3)where is the Euler-Mascheroni constant and is the th prime (Rosser and Schoenfeld 1962; Hardy and Wright 1979; Le Lionnais 1983; Ellison and Ellison 1985), or by the limit(4)According to Lindqvist and Peetre (1997), this was shown independently by Meisselin 1866 and Mertens (1874). Formula (3) is equivalent to(5)which follows from (4) using the Mercator series for with . is also given by the rapidly converging series(6)where is the Riemann zeta function, and is the Möbius..

The constant in Schnirelmann's theorem such that every integer is a sum of at most primes. Of course, by Vinogradov's theorem, it is known that 4 primes suffice for all sufficiently large numbers, but this constant gives a sufficient number for all numbers. The best current estimate is (Ramaré 1995), and a summary of progress on upper bounds for is summarized in the following table.author7Ramaré (1995)19Riesel and Vaughan (1983)26Deshouillers (1977)27Vaughan (1977)55Klimov (1975)115Klimov et al. (1972)159Deshouillers (1973)

Consider the Lagrange interpolatingpolynomial(1)through the points , where is the th prime. For the first few points, the polynomials are(2)(3)(4)(5)(6)So the first few values of , , , ..., are 2, 1, 1/2, , 1/8, , ... (OEIS A118210 and A118211).Now consider the partial sums of these coefficients, namely 2, 3, 7/2, 10/3, 83/24, 203/60, 2459/720, ... (OEIS A118203 and A118204). As first noted by F. Magata in 1998, the sum appears to converge to the value 3.407069... (OEIS A092894), now known as Magata's constant.

Sarnak's constant is the constant(1)(2)(OEIS A065476), where the product is over theodd primes.

Let be the smallest prime in the arithmetic progression for an integer . Letsuch that and . Then there exists a and an such that for all . is known as Linnik's constant.

Barban's constant is defined as(1)(2)(OEIS A175640), where the product is over the primes .

Let denote the number of primes of the form for , then(1)where is the logarithmic integral (Shanks 1960, pp. 321-332). Let denote the number of primes of the form for , then(2)(Shanks 1961, 1962). Let denote the number of pairs of primes and for , then(3)where(4)(Shanks 1960, pp. 201-203). Finally, let denote the number of pairs of primes and for , then(5)(Lal 1967), where is called Lal's constant. Shanks (1967) showed that .

The quadratic class number constant is a constant related to the average behavior of class numbers of real quadratic fields. It is given by(1)(2)(OEIS A065465), where the product is over the primes .

Just as the ratio of the arc length of a semicircle to its radius is always , the ratio of the arc length of the parabolic segment formed by the latus rectum of any parabola to its semilatus rectum (and focal parameter) is a universal constant(1)(2)(3)(4)(OEIS A103710). This can be seen from the equation of the arc length of a parabolic segment(5)by taking and plugging in and .The other conic sections, namely the ellipse and hyperbola, do not have such universal constants because the analogous ratios for them depend on their eccentricities. In other words, all circles are similar and all parabolas are similar, but the same is not true for ellipses or hyperbolas (Ogilvy 1990, p. 84).The area of the surface generated by revolving for about the -axis is given by(6)(7)(Love 1950, p. 288; OEIS A103713) and the area of the surface generated by revolving for about the -axis is(8)(9)(Love 1950, p. 288; OEIS A103714).The expected distance..

The silver ratio is the quantity defined by the continuedfraction(1)(2)(Wall 1948, p. 24). It follows that(3)so(4)(OEIS A014176).The sequence , of power fractional parts, where is the fractional part, is equidistributed for almost all real numbers , with the silver ratio being one exception.The more general expressions(5)are sometimes known in general as silver means (Knott). The first few values are summarized in the table below.OEISvalue1A0016221.618033988...2A0141762.414213562...3A0983163.302775637...4A0983174.236067977...5A0983185.192582403...

The silver constant is the algebraic number givenby(1)(2)(3)(OEIS A116425), where denotes a polynomial root.Defining the nested radical expression(4)the silver constant is given by(5)(T. Piezas, pers. comm., Feb. 16, 2006).The silver constant is the seventh Beraha constant. Surprisingly, it also appears in the 3-cycle of the logistic map.

Pogson's ratio is the constant(OEIS A189824) appearing in the definition of the astronomical magnitude (brightness) scale. This scale is based on the practice dating back to the Hellenistic Greeks of dividing the brightness of visible stars into six magnitudes, with the brightest stars being called first magnitude and the faintest visible stars sixth magnitude. Pogson (1856) systematized this system by defining a first magnitude star to be 100 times as bright as a sixth magnitude star using a logarithmic scale, thus making each magnitude times as bright as the previous one. This system remains widely adopted by astronomers today.

The plastic constant , sometimes also called the silver number or plastic number, is the limiting ratio of the successive terms of the Padovan sequence and Perrin sequence. It is given by(1)(2)(3)(OEIS A060006), where denotes a polynomial root. It is therefore an algebraic number of degree 3.It is also given by(4)where(5)where is the -function and the half-period ratio is equal to .The plastic constant was originally studied in 1924 by Gérard Cordonnier when he was 17. In his later correspondence with Dom Hans van der Laan, he described applications to architecture, using the name "radiant number." In 1958, Cordonnier gave a lecture tour that illustrated the use of the constant in many existing buildings and monuments (C. Mannu, pers comm., Mar. 11, 2006). satisfies the algebraic identities(6)and(7)and is therefore is one of the numbers for which there exist natural numbers and such that and . It was proven..

Let denote the square lattice with wraparound. Call an orientation of an assignment of a direction to each edge of , and denote the number of orientations of such that each vertex has two inwardly directed and two outwardly directly edges by . Such an orientation is said to obey the ice rule, or to consist of Eulerian orientation. For , 2, ..., the first few values of are 4, 18, 148, 2970, ... (OEIS A054759).Lieb showed that(1)(2)(3)(OEIS A118273; Finch 2003, p. 412), which is known as Lieb's square ice constant, also known as the square ice constant, residual entropy for square ice, and six-vertex entropy model.

The constant(OEIS A014715) giving the asymptotic rate of growth of the number of digits in the th term of the look and say sequence, given by the unique positive real root of the polynomialillustrated in the figure above. Note that the polynomialgiven in Conway (1987, p. 188) contains a misprint.The continued fraction for is 1, 3, 3, 2, 1, 2, 1, 5, 8, 4, 14, 3, 1, ... (OEIS A014967).

Pickover's sequence gives the starting positions in the decimal expansion of (ignoring the leading 3) in which the first digits of occur (counting the leading 2). So, since , the first digit "2" of occurs at position 6. Continuing, the sequence is given by 6, 28, 241, 11706, 28024, 33789, 1526800, 73154827, ... (OEIS A090898).Conversely, consider the sequence formed by the expansion of (ignoring the leading 2) in which the first digits of occur (counting the leading 3). So, since , the first digit "3" of occurs at position 17. Continuing, the sequence is given by 17, 189, 856, 17947, 53238, 1436935, 5000482, ... (OEIS A115234).

may be computed using a number of iterative algorithms. The best known such algorithms are the Archimedes algorithm, which was derived by Pfaff in 1800, and the Brent-Salamin formula. Borwein et al. (1989) discuss th-order iterative algorithms.The Brent-Salamin formula is a quadraticallyconverging algorithm.Another quadratically converging algorithm (Borweinand Borwein 1987, pp. 46-48) is obtained by defining(1)(2)and(3)(4)Then(5)with . decreases monotonically to with(6)for .A cubically converging algorithm which converges to the nearest multiple of to is the simple iteration(7)(Beeler et al. 1972). For example, applying to 23 gives the sequence 23, 22.1537796, 21.99186453, 21.99114858, ..., which converges to .A quartically converging algorithm is obtained by letting(8)(9)then defining(10)(11)Then(12)and converges to quartically with(13)(Borwein and Borwein 1987, pp. 170-171; Bailey 1988, Borwein..

The irrational constant(1)(2)(OEIS A060295), which is very close to an integer. Numbers such as the Ramanujan constant can be found using the theory of modular functions. In fact, the nine Heegner numbers (which include 163) share a deep number theoretic property related to some amazing properties of the j-function that leads to this sort of near-identity.Although Ramanujan (1913-1914) gave few rather spectacular examples of almost integers (such ), he did not actually mention the particular near-identity given above. In fact, Hermite (1859) observed this property of 163 long before Ramanujan's work. The name "Ramanujan's constant" was coined by Simon Plouffe and derives from an April Fool's joke played by Martin Gardner (Apr. 1975) on the readers of Scientific American. In his column, Gardner claimed that was exactly an integer, and that Ramanujan had conjectured this in his 1914 paper. Gardner admitted his hoax a..

Given a rectangle having sides in the ratio , the golden ratio is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio . Such a rectangle is called a golden rectangle. Euclid used the following construction to construct them. Draw the square , call the midpoint of , so that . Now draw the segment , which has length(1)and construct with this length. Now complete the rectangle , which is golden since(2)Successive points dividing a golden rectangle into squares lie on a logarithmic spiral (Wells 1991, p. 39; Livio 2002, p. 119) which is sometimes known as the golden spiral.The spiral is not actually tangent at these points, however, but passes through them and intersects the adjacent side, as illustrated above.If the top left corner of the original square is positioned at (0, 0), the center of the spiral occurs at the position(3)(4)(5)(6)(7)(8)(9)(10)(11)and..

A golden rhombus is a rhombus whose diagonals are in the ratio , where is the golden ratio.The faces of the acute golden rhombohedron, Bilinski dodecahedron, obtuse golden rhombohedron, rhombic hexecontahedron, and rhombic triacontahedron are golden rhombi.The half-angle is given by(1)(2)(3)(4)(OEIS A195693).Labeling the smaller interior angle as and the larger as , then(5)and(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(OEIS A105199 and A137218).The diagonal lengths of a golden rhombus with edge length are given by(18)(19)(20)(21)(22)(23)(24)(25)(OEIS A121570 and A179290),the inradius by(26)and the area by(27)

Let be an real square matrix with such that(1)for all real numbers , , ..., and , , ..., such that . Then Grothendieck showed that there exists a constant satisfying(2)for all vectors and in a Hilbert space with norms and . The Grothendieck constant is the smallest possible value of . For example, the best values known for small are(3)(4)(5)(Krivine 1977, 1979; König 1992; Finch 2003, p. 236).Now consider the limit(6)which is related to Khinchin's constant and sometimes also denoted . Krivine (1977) showed that(7)and postulated that(8)(OEIS A088367). The conjecture was refuted in 2011 by Yury Makarychev, Mark Braverman, Konstantin Makarychev, and Assaf Naor, who showed that is strictly less than Krivine's bound (Makarychev 2011).Similarly, if the numbers and and matrix are taken as complex, then a similar set of constants may be defined. These are known to satisfy(9)(10)(11)(Krivine 1977, 1979; König 1990, 1992; Finch..

The reciprocal of the arithmetic-geometric mean of 1 and ,(1)(2)(3)(4)(5)(6)(7)(OEIS A014549), where is the complete elliptic integral of the first kind, is a Jacobi theta function, and is the gamma function. This correspondence was first noticed by Gauss, and was the basis for his exploration of the lemniscate function (Borwein and Bailey 2003, pp. 13-15).Two rapidly converging series for are given by(8)(9)(Finch 2003, p. 421).Gauss's constant has continued fraction [0,1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, ...] (OEIS A053002).The inverse of Gauss's constant is given by(10)(OEIS A053004; Finch 2003, p. 420; Borwein and Bailey 2003, p. 13), which has [1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, ...] (OEIS A053003).The value(11)(OEIS A097057) is sometimes called the ubiquitous constant (Spanier and Oldham 1987; Schroeder 1994; Finch 2003, p. 421), and(12)(OEIS A076390) is sometimes called the secondlemniscate..

Jenny's constant is the name given (Munroe 2012) to the positive real constant defined by(1)(2)(OEIS A182369), the first few digits of which are 867-5309, corresponding to the fictitious phone number in the song "867-5309/Jenny" performed by Tommy Tutone in 1982.Other "simple" expressions that might vie for that moniker include(3)(4)(5)(6)(7)(8)(9)(10)where is the hard hexagon entropy constant, the first three of which are "better" than the canonical Jenny expression (E. Weisstein, Jul. 12, 2013).

Nice approximations for the golden ratio are given by(1)(2)the last of which is due to W. van Doorn (pers. comm., Jul. 18, 2006) and which are accurate to and , respectively. An even more amazing approximation uses Catalan's constant and the Feigenbaum constant is given by(3)which is accurate to within (D. Ross, cited in Pegg 2005).A curious (although not particularly useful) approximation due to D. Barron is given by(4)where is Catalan's constant and is the Euler-Mascheroni constant, which is good to two digits.

A curious approximation to the Feigenbaum constant is given by(1)where is Gelfond's constant, which is good to 6 digits to the right of the decimal point.M. Trott (pers. comm., May 6, 2008) noted(2)where is Gauss's constant, which is good to 4 decimal digits, and(3)where is the tetranacci constant, which is good to 3 decimal digits.A strange approximation good to five digits is given by the solution to(4)which is(5)where is the Lambert W-function (G. Deppe, pers. comm., Feb. 27, 2003).(6)gives to 3 digits (S. Plouffe, pers. comm., Apr. 10, 2006).M. Hudson (pers. comm., Nov. 20, 2004) gave(7)(8)(9)which are good to 17, 13, and 9 digits respectively.Stoschek gave the strange approximation(10)which is good to 9 digits.R. Phillips (pers. comm., Sept. 14, 2004-Jan. 25, 2005) gave the approximations(11)(12)(13)(14)(15)(16)where e is the base of the natural logarithm and..

Convergents of the pi continued fractions are the simplest approximants to . The first few are given by 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, ... (OEIS A002485 and A002486), which are good to 0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, ... (OEIS A114526) decimal digits, respectively.Two approximations follow from the near-identity function evaluated at and , giving(1)(2)which are good to 2 and 3 digits, respectively.Kochanski's approximation is the rootof(3)given by(4)which is good to 4 digits.Another curious fact is the almost integer(5)which can also be written as(6)(7)Here, is Gelfond's constant. Applying cosine a few more times gives(8)Another approximation involving is given by(9)which is good to 2 decimal digits (A. Povolotsky, pers. comm., Mar. 6, 2008).An apparently interesting near-identity is given by(10)which becomes less surprising when it is noted that 555555 is a repdigit,so the above is..

E. Pegg Jr. (pers. comm., Nov. 8, 2004) found an approximation to Apéry's constant given by(1)which is good to 6 digits.M. Hudson (pers. comm., Nov. 8, 2004) found the approximations(2)(3)(4)(5)(6)(7)where is the Euler-Mascheroni constant and is the golden ratio, which are good to 5, 7, 7, 8, 11, and 12 digits, respectively.A curious approximation to is given by(8)where is the Euler-Mascheroni constant, which is accurate to four digits (P. Galliani, pers. comm., April 19, 2002).Lima (2009) found the approximation(9)where is Catalan's constant, which is correct to 21 digits.

An amazing pandigital approximation to that is correct to 18457734525360901453873570 decimal digits is given by(1)found by R. Sabey in 2004 (Friedman 2004).Castellanos (1988ab) gives several curious approximations to ,(2)(3)(4)(5)(6)(7)which are good to 6, 7, 9, 10, 12, and 15 digits respectively.E. Pegg Jr. (pers. comm., Mar. 2, 2002), found(8)which is good to 7 digits.J. Lafont (pers. comm., MAy 16, 2008) found(9)where is a harmonic number, which is good to 7 digits.N. Davidson (pers. comm., Sept. 7, 2004) found(10)which is good to 6 digits.D. Barron noticed the curious approximation(11)where is Catalan's constant and is the Euler-Mascheroni constant, which however, is only good to 3 digits.

Approximations to Catalan's constant include(1)(2)(3)(4)(5)(6)(M. Hudson, pers. comm., Nov. 19, 2004), where is the golden ratio, which are good to 4, 5, 6, 6, 7, 7, and 9 digits, respectively.Other approximations include(7)(8)(K. Hammond, pers. comm., Dec. 31, 2005), where is the golden ratio, which are good to 5 and 9 digits, respectively.

The numerical value of is given by(OEIS A002392). It was computed to decimal digits by S. Kondo on May 20, 2011 (Yee).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 20, 111, 56, 9041, 4767, 674596, 24611354, 64653957, 131278082, ... (OEIS A228243).-constant primes occur at 1, 2, 40, 242, 842, 1541, 75067, ... decimal digits (OEIS A228240).The starting positions of the first occurrence of , 1, ... in the decimal expansion of (including the initial 2 and counting it as the first digit) are 3, 21, 1, 2, 13, 5, 17, 22, ... (OEIS A229197).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 22, 701, 7486, 88092, 1189434, 13426407, ... (OEIS A229124), which end at digits 7, 38, 351, 8493, 33058, 362945, ... (OEIS A229126).The digit strings 0123456789 first occurs starting at position 3349545080, but 9876543210 does not occur in the first..

The Earls sequence gives the starting position in the decimal digits of (or in general, any constant), not counting digits to the left of the decimal point, at which a string of copies of the number first occurs. The following table gives generalized Earls sequences for various constants, including .constantOEISsequenceApéry's constantA22907410, 57, 3938, 421, 41813, 1625571, 4903435, 99713909, ...Catalan's constantA2248192, 107, 1225, 596, 32187, 185043, 20444527, 92589355, 3487283621, ...Champernowne constantA2248961, 34, 56, 1222, 1555, 25554, 29998, 433330, 7988888882, 1101010101010, ...Copeland-Erdős constantA2248975, 113, 1181, 21670, 263423, 7815547, 35619942, 402720247, 450680638eA2248282, 252, 1361, 11806, 210482, 9030286, 3548262, 141850388, 1290227011Euler-Mascheroni constantA2248265, 139, 163, 10359, 86615, 193446, 236542, 6186099, 36151186Glaisher-Kinkelin constantA2257637,..

The decimal expansion of the natural logarithmof 2 is given by(OEIS A002162). It was computed to decimal digits by S. Kondo on May 14, 2011 (Yee).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 4, 419, 2114, 3929, 38451, 716837, 6180096, 10680693, 2539803904 (OEIS A228242).-constant primes occur at 321, 466, 1271, 15690, 18872, 89973, ... decimal digits (OEIS A228226).The starting positions of the first occurrence of , 1, ... in the decimal expansion of are 9, 4, 22, 3, 5, 10, 1, 6, 8, ... (OEIS A100077).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 2, 98, 604, 1155, 46847, 175403, ... (OEIS A036901), which end at digits 22, 444, 7655, 98370, 1107795, 12983306, ... (OEIS A036905).The digit string 0123456789 occurs starting at positions 3157027485, 8102152328, ... in the decimal digits of , and 9876543210 occurs starting..

The constant e with decimal expansion(OEIS A001113) can be computed to digits of precision in 10 CPU-minutes on modern hardware. was computed to digits by P. Demichel, and the first have been verified by X. Gourdon on Nov. 21, 1999 (Plouffe). was computed to decimal digits by S. Kondo on Jul. 5, 2010 (Yee).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 2, 252, 1361, 11806, 210482, 9030286, 3548262, 141850388, 1290227011, ... (OEIS A224828).The starting positions of the first occurrence of in the decimal expansion of (including the initial 2 and counting it as the first digit) are 14, 3, 1, 18, 11, 12, 21, 2, ... (OEIS A088576).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 6, 12, 548, 1769, 92994, ... (OEIS A036900), which end at digits 21, 372, 8092, 102128, ... (OEIS A036904).The digit sequence 0123456789..

Theodorus's constant has decimal expansion(OEIS A002194). It was computed to decimal digits by E. Weisstein on Jul. 23, 2013.The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 27, 215, 1651, 2279, 21640, 176497, 7728291, 77659477, 638679423, ... (OEIS A224874).-constant primes occur at 2, 3, 19, 111, 116, 641, 5411, 170657, ... (OEIS A119344) decimal digits.The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (including the initial 1 and counting it as the first digit) are 5, 1, 4, 3, 23, 6, 12, 2, 8, 18, ... (OEIS A229200).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 4, 91, 184, 5566, 86134, 35343, ... (OEIS A000000), which end at digits 23, 378, 7862, 77437, 1237533, 16362668, ... (OEIS A000000).The digit sequence 9876543210 does not occur in the first digits of , but 0123456789 does,..

An unsolved problem in mathematics attributed to Lehmer (1933) that concerns the minimum Mahler measure for a univariate polynomial that is not a product of cyclotomic polynomials. Lehmer (1933) conjectured that if is such an integer polynomial, then(1)(2)where , denoted by Lehmer (1933) and by Hironaka (2009), is the largest positive root of this polynomial. The roots of this polynomial, plotted in the left figure above, are very special, since 8 of the 10 lie on the unit circle in the complex plane. The roots of the polynomials (represented by half their coefficients) giving the two next smallest known Mahler measures are also illustrated above (Mossinghoff 1998, p. S11).The best current bound is that of Smyth (1971), who showed that , where is a nonzero nonreciprocal polynomial that is not a product of cyclotomic polynomials (Everest 1999), and is the real root of . Generalizations of Smyth's result have been constructed by Lloyd-Smith..

Ramanujan calculated (Hardy 1999, Le Lionnais 1983, Berndt 1994), while the correct value is(OEIS A070769; Derbyshire 2004, p. 114). The first decimal digits were computed by E. Weisstein on Oct. 7, 2013.-constant primes occur for 4, 144, 227, 444, 19474, ... (OEIS A122422) decimal digits.The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 3, 42, 178, 10013, 31567, 600035, 1253449, ... (OEIS A229071).The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (not including the initial 0 to the left of the decimal point) are 17, 1, 8, 5, 2, 3, 6, 34, 11, ... (OEIS A229201).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 7, 465, 102, 5858, 48441, ... (OEIS A000000), which end at digits 34, 512, 7454, 92508, 1414058, ... (OEIS A000000).The digit sequences 0123456789 and 9876543210 do not occur..

The numerical value of Khinchin's constant is given by(OEIS A002210). However, the numerical value of is notoriously difficult to calculate to high precision. Bailey et al. (1997) computed to 7350 digits, and the current record is digits, computed by Xavier Gourdon in 1997 with a computation requiring 22 hours and 23 minutes (Plouffe).The Earls sequence (starting position of copies of the digit ) for Khinchin's constant is given for , 2, ... by 9, 42, 1799, 494, 5760, ... (OEIS A224836), with the term being larger than .-constant primes occur at 1, 407, 878, 4443, 4981, 6551, 13386, 28433, ... decimal digits (OEIS A118327).The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (including the initial 2 and counting it as the first digit) are 8, 10, 1, 14, 5, 4, 2, 23, 3, 22, ... (OEIS A229196).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 7, 43,..

Let be a prime with digits and let be a constant. Call an "-prime" if the concatenation of the first digits of (ignoring the decimal point if one is present) give . Constant primes are therefore a special type of integer sequence primes, with e-primes, pi-primes, and phi-primes being perhaps the most prominent examples.The following table summarizes the indices of known constant primes for some named mathematical constants.constantname of primesOEIS giving primeApéry's constantA11933410, 55, 109, 141Catalan's constantA11832852, 276, 25477Champernowne constantA07162010, 14, 24, 235, 2804, 4347, 37735, 68433Copeland-Erdős constantA2275301, 2, 4, 11, 353, 355, 499, 1171, 1543, 5719, 11048ee-primeA0641181, 3, 7, 85, 1781, 2780, 112280, 155025Euler-Mascheroni constantA0658151, 3, 40, 185, 1038, 22610, 179849Glaisher-Kinkelin constantA1184207, 10, 18, 64, 71, 527, 1992, 5644, 8813, 19692Golomb-Dickman..

Scan the decimal expansion of a constant (including any digits to the left of the decimal point) until all -digit strings have been seen (including 0-padded strings). The following table then gives the number of digits that must be scanned to encounter all , 2, ...-digit strings (where "number of digits" means the ending-not starting-digit of an -digit string) together with the last -digit string encountered.constantOEISsequenceApéry's constantA03690623, 457, 7839, 83054, 1256587, 13881136, 166670757, ...A0369027, 89, 211, 2861, 43983, 29270, 8261623, ...Catalan's constantA00000032, 716, 7700, 86482, 1143572, ...A0000008, 45, 529, 2679, 24200, ...Champernowne constantA07229011, 192, 2893, 38894, 488895, 5888896, 68888897, 788888898, 8888888899, ...Copeland-Erdős constantA00000048, 934, 24437, 366399, 4910479, 49672582, ...A0000000, 84, 504, 8580, 07010, 088880, ...eA03690421, 372, 8092,..

A constant, sometimes also called a "mathematical constant," is any well-defined real number which is significantly interesting in some way. In this work, the term "constant" is generally reserved for real nonintegral numbers of interest, while "number" is used to refer to interesting integers (e.g., Brun's constant, but beast number). However, in contexts such as linear combination, the term "constant" is generally used to mean "scalar" or "real number," and need not exclude integer values.A function, equation, etc., is said to "be constant" (or be a constant function) if it always assumes the same value independent of how its parameters are varied.Certain constants are known to many decimal digits and recur throughout many diverse areas of mathematics, often in unexpected and surprising places (e.g., pi, e, and to some extent, the Euler-Mascheroni constant..

The Champernowne constant has decimal expansion(OEIS A033307).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 1, 34, 56, 1222, 1555, 25554, 29998, 433330, 7988888882, 1101010101010, ... (OEIS A224896).The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (not including the initial 0 to the left of the decimal point) are 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 1, ... (OEIS A229186).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 0, 00, 000, 0000, ..., which end at digits 11, 192, 2893, 38894, 488895, ... (OEIS A072290).The digit sequence 0123456789 first occurs at positions 11234567799, 22345677908, 33456779017, 44567790126, 55677901235, 66779012344, ... (OEIS A000000) and 9876543210 at positions 7777777779, 9876543212, 19987654323, 30998765434, 42099876545, 53209987656, 64320998767,..

Pythagoras's constant has decimal expansion(OEIS A000129), It was computed to decimal digits by A. J. Yee on Feb. 9, 2012.The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 2, 114, 1481, 3308, 72459, 226697, 969836, 119555442, 2971094743, ... (OEIS A224871).-constant primes occur at 55, 97, 225, 11260, 11540, ... (OEIS A115377) decimal digits.The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (including the initial 1 and counting it as the first digit) are 14, 1, 5, 7, 2, 8, 9, 12, 19, ... (OEIS A229199).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 8, 81, 748, 8505, 30103, 489568, ... (OEIS A000000), which end at digits 19, 420, 8326, 94388, 1256460, 13043524, ... (OEIS A000000).The digit sequence 9876543210 does not occur in the first digits of , but 0123456789 does, starting..

Based on methods developer in collaboration with M. Leclert, Catalan (1865) computed the constant(OEIS A006752) now known as Catalans' constant to 9 decimals. In 1867, M. Bresse subsequently computed to 24 decimal places using a technique from Kummer. Glaisher evaluated to 20 (Glaisher 1877) and subsequently 32 decimal digits (Glaisher 1913). Catalan's constant was computed to decimal digits by A. Roberts on Dec. 13, 2010 (Yee).The Earls sequence (starting position of copies of the digit ) for Catalan's constant is given for , 2, ... by 2, 107, 1225, 596, 32187, 185043, 20444527, 92589355, 3487283621, ... (OEIS A224819).-constant primes occur for 52, 276, 25477, ... (OEIS A118328) digits.It is not known if is normal, but the following table giving the counts of digits in the first terms shows that the decimal digits are very uniformly distributed up to at least .OEIS101000A22461506989769828996209997849998686999960671A2246162189410399832996971000293100038131000063052A22469601093980100781001681001789100051221000008063A22470607104101498599958099967299956761000014834A22471711110796110051100074100016599953771000018715A2247743108910031006210005399996599993091000007776A22477511278985998610020199871210000674999988167A2248160111241032100281000831000510100038631000005768A2248170310210581019210035299929899974371000008639A224818312111952100841001729998121000004399992436The..

The golden ratio has decimal expansion(OEIS A001622). It can be computed to digits of precision in 24 CPU-minutes on modern hardware and was computed to decimal digits by A. J. Yee on Jul. 8, 2010.The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 2, 62, 158, 1216, 72618, 2905357, 7446157, 41398949, 1574998166, ... (OEIS A224844).The digit sequence 0123456789 does not occur in the first digits of , but 9876543210 does, starting at position (E. Weisstein, Jul. 22, 2013).Phi-primes, i.e., -constant primes occur for 7, 13, 255, 280, 97241, ... (OEIS A064119) decimal digits.The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (including the initial 1 and counting it as the first digit) are 5, 1, 20, 6, 12, 23, 2, 11, 4, 8, 232, ... (OEIS A088577).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit..

Apéry's constant is defined by(OEIS A002117) where is the Riemann zeta function. was computed to decimal digits by E. Weisstein on Sep. 16, 2013.The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 10, 57, 3938, 421, 41813, 1625571, 4903435, 99713909, ... (OEIS A229074).-constant prime occur for , 55, 109, 141, ... (OEIS A119334), corresponding to the primes 1202056903, 1202056903159594285399738161511449990764986292340498881, ... (OEIS A119333).The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (not including the initial 0 to the left of the decimal point) are 3, 1, 2, 10, 16, 6, 7, 23, 18, 8, ... (OEIS A229187).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 7, 89, 211, 2861, 43983, 292702, 8261623, ... (OEIS A036902), which end at digits 23, 457, 7839, 83054, 1256587,..

Apéry's constant is defined by(1)(OEIS A002117) where is the Riemann zeta function. Apéry (1979) proved that is irrational, although it is not known if it is transcendental. Sorokin (1994) and Nesterenko (1996) subsequently constructed independent proofs for the irrationality of (Hata 2000). arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics.The following table summarizes progress in computing upper bounds on the irrationality measure for . Here, the exact values for is given by(2)(3)(Hata 2000).upper boundreference15.513891Rhin and Viola (2001)28.830284Hata (1990)312.74359Dvornicich and Viola (1987)413.41782Apéry (1979), Sorokin (1994), Nesterenko (1996), Prévost (1996)Beukers (1979) reproduced Apéry's rational approximation to using the triple..

The continued fraction for is given by [1; 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, ...] (OEIS A099803).The positions at which the numbers 1, 2, ... occur in the continued fraction are0, 1, 6, 2, 47, 28, 21, 107, 114, ... (OEIS A000000).The high-water marks are 1, 2, 4, 47, 99, 294, 527, 616, 1152, ... (OEIS A099804), which occur at positions 0, 1, 2, 11, 69, 125, 201, 584, 1591, 2435, ... (OEIS A229230).Let the continued fraction of be denoted and let the denominators of the convergents be denoted , , ..., . Then plots above show successive values of , , , which appear to converge to Khinchin's constant (left figure) and , which appear to converge to the Lévy constant (right figure), although neither of these limits has been rigorously established.

The simple continued fraction representations of given by [2; 1, 2, 1, 1, 4, 1, 1, 6, ...] (OEIS A003417). This continued fraction is sometimes known as Euler's continued fraction. A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.The convergents can be given in closed form as ratios of confluent hypergeometric functions of the first kind (Komatsu 2007ab), with the first few being 2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, ... (OEIS A007676 and A007677). These are good to 0, 0, 1, 1, 2, 3, 3, 4, 5, 5, ... (OEIS A114539) decimal digits, respectively.Other continued fraction representations are(1)(2)(3)(Olds 1963, pp. 135-136). Amazingly, not only the continued fractions , but those of rational powers of show regularity, for example(4)(5)(6)(7)A beautiful non-simple continued fraction for is given by(8)(Wall 1948, p. 348).Let the continued fraction of be denoted..

The simple continued fraction for pi is given by [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...] (OEIS A001203). A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.The first few convergents are 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, ... (OEIS A002485 and A002486), which are good to 0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, ... (OEIS A114526) decimal digits, respectively.The very large term 292 means that the convergent(1)is an extremely good approximation good to six decimal places that was first discovered by astronomer Tsu Ch'ung-Chih in the fifth century A.D. (Gardner 1966, pp. 91-102). A nice expression for the third convergent of is given by(2)(Stoschek).The Engel expansion of is 1, 1, 1, 8, 8, 17, 19, 300, 1991, 2492, ... (OEIS A006784).The following table summarizes some record computations of the continued fraction of pi.termsdatereference1977W. Gosper..

The continued fraction for is [2; 1, 2, 5, 1, 1, 2, 1, 1, ...] (OEIS A002211). A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.The convergents are 2, 3, 8/3, 43/16, 51/19, ... (OEIS A127005and A127006).The incrementally largest terms are 2, 5, 10, 24, 90, 770, ... (OEIS A054866), which occur at positions 0, 3, 10, 15, 23, 104, 1701, ... (OEIS A224852; illustrated above).The plot above shows the positions of the first occurrences of 1, 2, 3, ... in the continued fraction, the first few of which are 1, 0, 9, 46, 3, 33, 75, 64, 118, 10, 103, 26, 102, 109, ... (OEIS A224851). The smallest number not occurring in the first terms of the continued fraction are 236, 260, 265, 279, 282, ... (E. Weisstein, Jul. 22, 2013).

Let(1)be the simple continued fraction of a "generic" real number , where the numbers are the partial quotients. Khinchin (1934) considered the limit of the geometric mean(2)as . Amazingly, except for a set of measure 0, this limit is a constant independent of given by(3)(OEIS A002210), as proved in Kac (1959).The constant is known as Khinchin's constant, and is commonly also spelled "Khintchine'sconstant" (Shanks and Wrench 1959, Bailey et al. 1997).It is implemented as Khinchin, where its value is cached to 1100-digit precision. However, the numerical value of is notoriously difficult to calculate to high precision, so computation of more digits get increasingly slower.It is not known if is irrational, let alone transcendental.While it is known that almost all numbers have limits that approach , this fact has not been proven for any explicit real number , e.g., a real number cast in terms of fundamental constants..

The first few terms in the continued fraction of the Champernowne constant are [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 45754...10987, 6, 1, 1, 21, ...] (OEIS A030167), and the number of decimal digits in these terms are 0, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 166, 1, ... (OEIS A143532). E. W. Weisstein computed terms of the continued fraction on Jun. 30, 2013 using the Wolfram Language.First occurrences of the terms 1, 2, 3, ... in the continued fraction occur at , 28, 13, 9, 93, 20, 31, 2, 3, 339, 71, 126, 107, ... (OEIS A038706). The smallest unknown value is 188, which has .The continued fraction contains sporadic very large terms, making the continued fraction difficult to calculate. However, the size of the continued fraction high-water marks display apparent patterns (Sikora 2012). Large terms greater than occur at positions 5, 19, 41, 102, 163, 247, 358, 460, ... and have 6, 166, 2504, 140, 33102, 109, 2468,..

The simple continued fraction representations for Catalan's constant is [0, 1, 10, 1, 8, 1, 88, 4, 1, 1, ...] (OEIS A014538). A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.Record computations are summarized below.termsdatebyJul. 20, 2013E. WeissteinAug. 7, 2013E. WeissteinThe plot above shows the positions of the first occurrences of 1, 2, 3, ... in the continued fraction, the first few of which are 1, 13, 14, 7, 45, 36, 10, 4, 21, 2, ... (OEIS A196461; illustrated above). The smallest number not occurring in the first terms of the continued fraction are 31516, 31591, 32600, 32806, 33410, ... (E. Weisstein, Aug. 8, 2013).The cumulative largest terms in the continued fraction are 0, 1, 10, 88, 322, 330, 1102, 6328, ... (OEIS A099789), which occur at positions 0, 1, 2, 6, 105, 284, 747, 984, 2230, 5377, ... (OEIS A099790).Let the continued fraction..

The continued fraction for Apéry's constant is [1; 4, 1, 18, 1, 1, 1, 4, 1, ...] (OEIS A013631).The positions at which the numbers 1, 2, ... occur in the continued fraction are 0, 11, 24, 1, 63, 26, 16, 139, 9, 118, 20, ... (OEIS A229057). The incrementally maximal terms are 1, 4, 18, 30, 428, 458, 527, ... (OEIS A033166), which occur at positions 0, 1, 3, 28, 62, 571, 1555, 2012, 2529, ... (OEIS A229055).Let the continued fraction of be denoted and let the denominators of the convergents be denoted , , ..., . Then plots above show successive values of , , , which appear to converge to Khinchin's constant (left figure) and , which appear to converge to the Lévy constant (right figure), although neither of these limits has been rigorously established.

A numberwhere is an integer or rational number, is the inverse tangent, and is the inverse cotangent. Gregory numbers arise in the determination of Machin-like formulas. Every Gregory number can be expressed uniquely as a linear combination of s where the s are Størmer numbers.

Lehmer (1938) showed that every positive irrational number has a unique infinite continued cotangent representation of the form(1)where the s are nonnegative and(2)Note that this growth condition on coefficients is essential for the uniqueness of Lehmer expansion.The following table summarizes the coefficients for various special constants.OEISeA0026682, 8, 75, 8949, 119646723, 15849841722437093, ...Euler-Mascheroni constant A0817820, 1, 3, 16, 389, 479403, 590817544217, ...golden ratio A0062671, 4, 76, 439204, 84722519070079276, ...Lehmer's constant A0020650, 1, 3, 13, 183, 33673, ...A0026673, 73, 8599, 400091364,371853741549033970, ...Pythagoras's constant A0026661, 5, 36, 3406, 14694817,727050997716715, ...The expansion for the golden ratio has the beautiful closed form(3)where is a Lucas number.An illustration of a different cotangent expansion for that is not a Lehmer expansion because its coefficients..

The Lehmer cotangent expansion for whichthe convergence is slowest occurs when the inequality in the recurrence equation(1)for(2)is replaced by equality, giving and(3)for .This recurrences gives values of corresponding to 0, 1, 3, 13, 183, 33673, ... (OEIS A002065), and defines the constant known as Lehmer's constant as(4)(5)(6)(OEIS A030125). is not an algebraic number of degree less than 4, but Lehmer's approach cannot show whether is transcendental.

The logarithmic integral is defined as theCauchy principal value(1)(2)Soldner's constant, denoted (or sometimes ) is the root of the logarithmic integral,(3)so that(4)for (Soldner 1812; Nielsen 1965, p. 88). Ramanujan calculated (Hardy 1999, Le Lionnais 1983, Berndt 1994), while the correct value is 1.45136923488... (OEIS A070769; Derbyshire 2004, p. 114).

The decimal expansion of the natural logarithm of 10 is given by(1)(OEIS A002392).It is also given by the BBP-type formulas(2)(3)(4)(5)(6)(7)(E. W. Weisstein, Aug. 28, 2008).

Expanding the Riemann zeta function about gives(1)(Havil 2003, p. 118), where the constants(2)are known as Stieltjes constants.Another sum that can be used to define the constants is(3)These constants are returned by the WolframLanguage function StieltjesGamma[n].A generalization takes as the coefficient of is the Laurent series of the Hurwitz zeta function about . These generalized Stieltjes constants are implemented in the Wolfram Language as StieltjesGamma[n, a].The case gives the usual Euler-Mascheroni constant(4)A limit formula for is given by(5)where is the imaginary part and is the Riemann zeta function.An alternative definition is given by absorbing the coefficient of into the constant,(6)(e.g., Hardy 1912, Kluyver 1927).The Stieltjes constants are also given by(7)Plots of the values of the Stieltjes constants as a function of are illustrated above (Kreminski). The first few numerical values are given in the..

There are two sets of constants that are commonly known as Lebesgue constants. The first is related to approximation of function via Fourier series, which the other arises in the computation of Lagrange interpolating polynomials.Assume a function is integrable over the interval and is the th partial sum of the Fourier series of , so that(1)(2)and(3)If(4)for all , then(5)and is the smallest possible constant for which this holds for all continuous . The first few values of are(6)(7)(8)(9)(10)(11)(12)(13)Some sum formulas for include(14)(15)(Zygmund 1959) and integral formulas include(16)(17)(Hardy 1942). For large ,(18)This result can be generalized for an -differentiable function satisfying(19)for all . In this case,(20)where(21)(Kolmogorov 1935, Zygmund 1959).Watson (1930) showed that(22)where(23)(24)(25)(OEIS A086052), where is the gamma function, is the Dirichlet lambda function, and is the Euler-Mascheroni constant.Define..

Wyler's constant is defined as(1)(2)(3)(Wyler 1969, 1971; OEIS A180872 and A180873), which at the time it was proposed, agreed with experiment to within ppm for the value of the fine structure constant in physics. The current best value for is given by(4)(Hanneke et al. 2008).While it appears to have a connection with the invariance group of a relativistic quantum theoretical wave equation, a number of errors in Wyler's papers are cited in Robertson (1971). Robertson (1971) also notes, "It is appealing to think that it [the fine-structure constant] might be derivable theoretically. Wyler's number ... appears to have better chances to be derived from a theory than any of the other numbers that also agree with experiment. It may be that even though the expression (8) [from Wyler's paper] is not correct, the number (12) [Wyler's constant] somehow is correct." Adler (1972) terms the constant, "a number in search of a theory"..

The White House switchboard constant is the name given by Munroe (2012) to the constant(1)(2)(OEIS A182064), the first few digits of which are 202-456-1414, which is the phone number of the switchboard of the White House (home of U.S. President).Other "simple" expressions that might vie for that moniker include(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)where is the Euler-Mascheroni constant, all of which are "better" than the canonical White House switchboard expression (E. Weisstein, Jul. 13, 2013).

There are (at least) two mathematical constants associated with Theodorus. The first Theodorus's constant is the elementary algebraic number , i.e., the square root of 3. It has decimal expansion(1)(OEIS A002194) and is named after Theodorus, who proved that the square roots of the integers from 3 to 17 (excluding squares 4, 9,and 16) are irrational (Wells 1986, p. 34). The space diagonal of a unit cube has length . has continued fraction [1, 1, 2, 1, 2, 1, 2, ...] (OEIS A040001). In binary, it is represented by(2)(OEIS A004547).Another constant sometimes known as the constant of Theodorus is the slope of a continuous analog of the discrete Theodorus spiral due to Davis (1993) at the point , given by(3)(4)(5)(6)(OEIS A226317; Finch 2009), where is the Riemann zeta function.

In this work, the name Pythagoras's constant will be given to the squareroot of 2,(1)(OEIS A002193), which the Pythagoreans provedto be irrational.In particular, is the length of the hypotenuse of an isosceles right triangle with legs of length one, and the statement that it is irrational means that it cannot be expressed as a ratio of integers and . Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality of while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pythagoreans. A slight generalization is sometimes known as Pythagoras's theorem.Theodorus subsequently proved that the square roots of the numbers from 3 to 17 (excluding 4, 9,and 16) are also irrational (Wells 1986, p. 34).It is not known if Pythagoras's constant is normalto any base (Stoneham 1970, Bailey and Crandall 2003).The continued fraction for..

Closed forms are known for the sums of reciprocals of even-indexed Lucasnumbers(1)(2)(3)(4)(5)(OEIS A153415), where is the golden ratio, is a q-polygamma function, and is a Jacobi theta function, and odd-indexed Lucas numbers(6)(7)(8)(9)(10)(11)(OEIS A153416), where is a Lambert series (Borwein and Borwein 1987, pp. 91-92). This gives the reciprocal Lucas constant as(12)(13)(14)(15)(16)(OEIS A093540), where is the golden ratio and is a Fibonacci number.Borwein and Borwein (1987, pp. 94-101) give a number of related beautiful formulas.

Closed forms are known for the sums of reciprocals of even-indexed Fibonaccinumbers(1)(2)(3)(4)(5)(6)(7)(OEIS A153386; Knopp 1990, Ch. 8, Ex. 114; Paszkowski 1997; Horadam 1988; Finch 2003, p. 358; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is the golden ratio, is a q-polygamma function, and is a Lambert series (Borwein and Borwein 1987, pp. 91 and 95) and odd-indexed Fibonacci numbers(8)(9)(10)(11)(12)(13)(OEIS A153387; Landau 1899; Borwein and Borwein 1997, p. 94; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is a Jacobi elliptic function. Together, these give a closed form for the reciprocal Fibonacci constant of(14)(15)(16)(17)(18)(OEIS A079586; Horadam 1988; Griffin 1992; Zhao 1999; Finch 2003, p. 358). The question of the irrationality of was formally raised by Paul Erdős and this sum was proved to be irrational by André-Jeannin (1989).Borwein..

The golden ratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagon, pentagram, decagon and dodecahedron. It is denoted , or sometimes .The designations "phi" (for the golden ratio conjugate ) and "Phi" (for the larger quantity ) are sometimes also used (Knott), although this usage is not necessarily recommended.The term "golden section" (in German, goldener Schnitt or der goldene Schnitt) seems to first have been used by Martin Ohm in the 1835 2nd edition of his textbook Die Reine Elementar-Mathematik (Livio 2002, p. 6). The first known use of this term in English is in James Sulley's 1875 article on aesthetics in the 9th edition of the Encyclopedia Britannica. The symbol ("phi") was apparently first used by Mark Barr at the beginning of the 20th century in commemoration..

A short mnemonic for remembering the first seven decimal digits of is "How I wish I could calculate pi" (C. Heckman, pers. comm., Feb. 3, 2005). Eight digits are given by "May I have a large container of coffee?" giving 3.1415926 (Gardner 1959; 1966, p. 92; Eves 1990, p. 122, Davis 1993, p. 9). "But I must a while endeavour to reckon right" gives nine correct digits (3.14159265). "May I have a white telephone, or pastel color" (M. Amling, pers. comm., Jul. 31, 2004) also gives nine correct digits.A more substantial mnemonic giving 15 digits (3.14159265358979) is "How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics," originally due to Sir James Jeans (Gardner 1966, p. 92; Castellanos 1988, p. 152; Eves 1990, p. 122; Davis 1993, p. 9; Blatner 1997, p. 112). A slight extension..

The constant is base of the natural logarithm. is sometimes known as Napier's constant, although its symbol () honors Euler. is the unique number with the property that the area of the region bounded by the hyperbola , the x-axis, and the vertical lines and is 1. In other words,(1)With the possible exception of , is the most important constant in mathematics since it appears in myriad mathematical contexts involving limits and derivatives. The numerical value of is(2)(OEIS A001113). can be defined by the limit(3)(illustrated above), or by the infinite series(4)as first published by Newton (1669; reprinted in Whiteside 1968, p. 225). is given by the unusual limit(5)(Brothers and Knox 1998).Euler (1737; Sandifer 2006) proved that is irrational by proving that has an infinite simple continued fraction (; Nagell 1951), and Liouville proved in 1844 that does not satisfy any quadratic equation with integral coefficients (i.e., if it is..

The natural logarithm of 2 is a transcendental quantity that arises often in decay problems, especially when half-lives are being converted to decay constants. has numerical value(1)(OEIS A002162).The irrationality measure of is known to be less than 3.8913998 (Rukhadze 1987, Hata 1990).It is not known if is normal (Bailey and Crandall 2002).The alternating series and BBP-typeformula(2)converges to the natural logarithm of 2, where is the Dirichlet eta function. This identity follows immediately from setting in the Mercator series, yielding(3)It is also a special case of the identity(4)where is the Lerch transcendent.This is the simplest in an infinite class of such identities, the first few of which are(5)(6)(E. W. Weisstein, Oct. 7, 2007).There are many other classes of BBP-type formulas for , including(7)(8)(9)(10)(11)Plouffe (2006) found the beautiful sum(12)A rapidly converging Zeilberger-type sum..

There is a series of BBP-type formulas for in powers of ,(1)(2)(3)(4)(5)(6),(7)(8)(9)(10)some of which are noted by Bailey et al. (1997), and ,(11)(12)Another identity is(13)where is the polylogarithm. (13) is equivalent to(14)(Bailey et al. 1997).

Catalan's constant is a constant that commonly appears in estimates of combinatorial functions and in certain classes of sums and definite integrals. It is usually denoted (this work), (e.g., Borwein et al. 2004, p. 49), or (Wolfram Language).Catalan's constant may be defined by(1)(Glaisher 1877, who however did not explicitly identify the constant in this paper). It is not known if is irrational.Catalan's constant is implemented in the WolframLanguage as Catalan.The constant is named in honor of E. C. Catalan (1814-1894), who first gave an equivalent series and expressions in terms of integrals. Numerically,(2)(OEIS A006752). can be given analytically by the following expressions(3)(4)(5)where is the Dirichlet beta function, is Legendre's chi-function, is the Glaisher-Kinkelin constant, and is the partial derivative of the Hurwitz zeta function with respect to the first argument.Glaisher (1913) gave(6)(Vardi..

There are many formulas of of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. is intimately related to the properties of circles and spheres. For a circle of radius , the circumference and area are given by(1)(2)Similarly, for a sphere of radius , the surface area and volume enclosed are(3)(4)An exact formula for in terms of the inverse tangents of unit fractions is Machin's formula(5)There are three other Machin-like formulas,as well as thousands of other similar formulas having more terms.Gregory and Leibniz found(6)(7)(Wells 1986, p. 50), which is known as the Gregory series and may be obtained by plugging into the Leibniz series for . The error after the th term of this series in the Gregory series is larger than so this sum converges so slowly that 300 terms are not sufficient to calculate correctly to two decimal places! However, it can be transformed..

The Gregory series is a pi formula found by Gregory and Leibniz and obtained by plugging into the Leibniz series,(1)(Wells 1986, p. 50). The formula converges very slowly, but its convergence can be accelerated using certain transformations, in particular(2)where is the Riemann zeta function (Vardi 1991).Taking the partial series gives the analytic result(3)Rather amazingly, expanding about infinity gives the series(4)(Borwein and Bailey 2003, p. 50), where is an Euler number. This means that truncating the Gregory series at half a large power of 10 can give a decimal expansion for whose decimal digits are largely correct, but where wrong digits occur with precise regularity. For example, taking gives where the sequence of differences is precisely twice the Euler (secant) numbers. In fact, just this pattern of digits was observed by J. R. North in 1988 before the closed form of the truncated series was known..

has decimal expansion given by(1)(OEIS A000796). The following table summarizes some record computations of the digits of .1999Kanada, Ushio and KurodaDec. 2002Kanada, Ushio and Kuroda (Peterson 2002, Kanada 2003)Aug. 2012A. J. Yee (Yee)Aug. 2012S. Kondo and A. J. Yee (Yee)Dec. 2013A. J. Yee and S. Kondo (Yee)The calculation of the digits of has occupied mathematicians since the day of the Rhind papyrus (1500 BC). Ludolph van Ceulen spent much of his life calculating to 35 places. Although he did not live to publish his result, it was inscribed on his gravestone. Wells (1986, p. 48) discusses a number of other calculations. The calculation of also figures in the Season 2 Star Trek episode "Wolf in the Fold" (1967), in which Captain Kirk and Mr. Spock force an evil entity (composed of pure energy and which feeds on fear) out of the starship..

The BBP (named after Bailey-Borwein-Plouffe) is a formula for calculating pidiscovered by Simon Plouffe in 1995,Amazingly, this formula is a digit-extraction algorithm for in base 16.Following the discovery of this and related formulas, similar formulas in other bases were investigated. This class of formulas are now known as BBP-type formulas.

The constant pi, denoted , is a real number defined as the ratio of a circle's circumference to its diameter ,(1)(2) has decimal expansion given by(3)(OEIS A000796). Pi's digits have many interesting properties, although not very much is known about their analytic properties. However, spigot (Rabinowitz and Wagon 1995; Arndt and Haenel 2001; Borwein and Bailey 2003, pp. 140-141) and digit-extraction algorithms (the BBP formula) are known for .A brief history of notation for pi is given by Castellanos (1988ab). is sometimes known as Archimedes' constant or Ludolph's constant after Ludolph van Ceulen (1539-1610), a Dutch calculator. The symbol was first used by Welsh mathematician William Jones in 1706, and subsequently adopted by Euler. In Measurement of a Circle, Archimedes (ca. 225 BC) obtained the first rigorous approximation by inscribing and circumscribing -gons on a circle using the Archimedes algorithm. Using (a 96-gon),..

For , the Riemann zeta function is given by(1)(2)where is the th prime. This is Euler's product (Whittaker and Watson 1990), called by Havil (2003, p. 61) the "all-important formula" and by Derbyshire (2004, pp. 104-106) the "golden key."This can be proved by expanding the product, writing each term as a geometricseries, expanding, multiplying, and rearranging terms,(3)Here, the rearrangement leading to equation (1) follows from the fundamental theorem of arithmetic, since each product of prime powers appears in exactly one denominator and each positive integer equals exactly one product of prime powers.This product is related to the Möbius function via(4)which can be seen by expanding the product to obtain(5)(6)(7)(8)(9), but the finite product exists, giving(10)For upper limits , 1, 2, ..., the products are given by 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, ... (OEIS A060753 and..

Let be a complex number, then inequality(1)holds in the lens-shaped region illustrated above. Written explicitly in terms of real variables, this can be written as(2)where(3)The area enclosed is roughly(4)(OEIS A140133).This region can be parameterized in terms of a variable as(5)(6)Written parametrically in terms of the Cartesian coordinates,(7)(8)This region is intimately related to the study of Bessel functions and Kapteynseries (Plummer 1960, p. 47; Watson 1966, p. 270). reaches its maximum value at (OEIS A085984; Goursat 1959, p. 120; Le Lionnais 1983, p. 36), given by the root of(9)or equivalently by the root of(10)as noted by Stieltjes.The minimum value of corresponding to the maximum value is (OEIS A033259; Plummer 1960, p. 47; Watson 1966, p. 270), which is known as the Laplace limit constant. It is precisely the point at which Laplace's formula for solving Kepler's equation begins..

Porter's constant is the constant appearing in formulasfor the efficiency of the Euclidean algorithm,(1)(2)(3)(OEIS A086237), where is the Euler-Mascheroni constant, is the Riemann zeta function, and is the Glaisher-Kinkelin constant (Knuth 1998, p. 357). The notation is generally used for this constant (Knuth 1998, p. 357, Finch 2003, pp. 156-157), though other authors use (Ustinov 2010) or (Dimitrov et al. 2000).The related constant originally considered by Porter (1975) and Knuth (1976) was denoted and , respectively, and defined by(4)(5)Knuth (1976) suggested be called the Lochs-Porter constant due to the work of Lochs (1961).

The golden angle is the angle that divides a full angle in a golden ratio (but measured in the opposite direction so that it measures less than ), i.e.,(1)(2)(3)(4)(5)(6)(7)(OEIS A131988 and A096627;Livio 2002, p. 112).It is implemented in the Wolfram Languageas GoldenAngle.van Iterson showed in 1907 that points separated by on a tightly bound spiral tends to produce interlocked spirals winding in opposite directions, and that the number of spirals in these two families tend to be consecutive Fibonacci numbers (Livio 2002, p. 112).Another angle related to the golden ratio is theangle(8)or twice this angle(9)the later of which is the smaller interior angle in the goldenrhombus.

The approximation for pi given by(1)(2)(3)In the above figure, let , and construct the circle centered at of radius 1. This intersects at point . Now construct the circle about with radius 1. The circles and intersect in , and the line intersects the perpendicular to through in the point . Now construct the point to be a distance 3 along . The line segment is then of length(4)This construction was given by the Polish Jesuit priest Kochansky (Steinhaus 1999).

Mergelyan's theorem can be stated as follows (Krantz 1999). Let be compact and suppose has only finitely many connected components. If is holomorphic on the interior of and if , then there is a rational function with poles in such that(1)A consequence is that if is an infinite set of disjoint open disks of radius such that the union is almost the unit disk. Then(2)Define(3)Then there is a number such that diverges for and converges for . The above theorem gives(4)There exists a constant which improves the inequality, and the best value known is(5)

Let be the set of complex analytic functions defined on an open region containing the closure of the unit disk satisfying and . For each in , let be the supremum of all numbers such that contains a disk of radius . ThenThis constant is called the Landau constant, or the Bloch-Landau constant. Robinson (1938, unpublished) and Rademacher (1943) derived the bounds(OEIS A081760), where is the gamma function, and conjectured that the second inequality is actually an equality.

Define the sequence , , and(1)for . The first few values are(2)(3)(4)(5)Janssen and Tjaden (1987) showed that this sequence converges for exactly one value , where (OEIS A085835), confirming Grossman's conjecture. However, no analytic form is known for this constant, either as the root of a function or as a combination of other constants. The plot above shows the first few iterations of for to 30, with odd shown in red and even shown in blue, for ranging from 0 to 1. As can be seen, the solutions alternate by parity. For each fixed , the red values go to 0, while the blue values go to some positive number.Nyerges (2000) has generalized the recurrence to the functional equation(6)

The de Bruijn constant, also called the Copson-de Bruijn constant, is the minimal constant(OEIS A113276) such that the inequalityalways holds.

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