Let a convex polygon be inscribed in a circle and divided into triangles from diagonals from one polygon vertex. The sum of the radii of the circles inscribed in these triangles is the same independent of the polygon vertex chosen (Johnson 1929, p. 193).If a triangle is inscribed in a circle, another circle inside the triangle, a square inside the circle, another circle inside the square, and so on. Then the equation relating the inradius and circumradius of a regular polygon,(1)gives the ratio of the radii of the final to initial circles as(2)Numerically,(3)(OEIS A085365), where is the corresponding constant for polygon circumscribing. This constant is termed the Kepler-Bouwkamp constant by Finch (2003). Kasner and Newman's (1989) assertion that is incorrect, as is the value of 0.8700... given by Prudnikov et al. (1986, p. 757)...
Circumscribe a triangle about a circle, another circle around the triangle, a square outside the circle, another circle outside the square, and so on. The circumradius and inradius for an -gon are then related by(1)so an infinitely nested set of circumscribed polygons and circles has(2)(3)(4)Kasner and Newman (1989) and Haber (1964) state that , but this is incorrect, and the actual answer is(5)(OEIS A051762).By writing(6)it is possible to expand the series about infinity, change the order of summation, do the sum symbolically, and obtain the quickly converging series(7)where is the Riemann zeta function.Bouwkamp (1965) produced the following infinite productformulas for the constant,(8)(9)(10)where is the sinc function (cf. Prudnikov et al. 1986, p. 757), is the Riemann zeta function, and is the Dirichlet lambda function. Bouwkamp (1965) also produced the formula with accelerated convergence(11)where(12)(cited in Pickover..
Let be the matrix whose th entry is 1 if divides and 0 otherwise, let be the diagonal matrix , where is the totient function, and let be the matrix whose th entry is the greatest common divisor . Then Le Paige's theorem states thatwhere denotes the transpose (Le Paige 1878, Johnson 2003).As a corollary,(Smith 1876, Johnson 2003). For , 2, ... the first few values are 1, 1, 2, 4, 16, 32, 192, 768, ... (OEIS A001088).
The -hypersphere (often simply called the -sphere) is a generalization of the circle (called by geometers the 2-sphere) and usual sphere (called by geometers the 3-sphere) to dimensions . The -sphere is therefore defined (again, to a geometer; see below) as the set of -tuples of points (, , ..., ) such that(1)where is the radius of the hypersphere.Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "-sphere," with geometers referring to the number of coordinates in the underlying space ("thus a two-dimensional sphere is a circle," Coxeter 1973, p. 125) and topologists referring to the dimension of the surface itself ("the -dimensional sphere is defined to be the set of all points in satisfying ," Hocking and Young 1988, p. 17; "the -sphere is ," Maunder 1997, p. 21). A geometer would therefore regard the object described by(2)as a 2-sphere,..
A portion of a disk whose upper boundary is a (circular) arc and whose lower boundary is a chord making a central angle radians (), illustrated above as the shaded region. The entire wedge-shaped area is known as a circular sector.Circular segments are implemented in the Wolfram Language as DiskSegment[x, y, r, q1, q2]. Elliptical segments are similarly implemented as DiskSegment[x, y, r1, r2, q1, q2].Let be the radius of the circle, the chord length, the arc length, the height of the arced portion, and the height of the triangular portion. Then the radius is(1)the arc length is(2)the height is(3)(4)(5)and the length of the chord is(6)(7)(8)(9)From elementary trigonometry, the angle obeys the relationships(10)(11)(12)(13)The area of the (shaded) segment is then simply given by the area of the circular sector (the entire wedge-shaped portion) minus the area of the bottom triangular portion,(14)Plugging in gives(15)(16)(17)(18)where..
A Chaitin's constant, also called a Chaitin omega number, introduced by Chaitin (1975), is the halting probability of a universal prefix-free (self-delimiting) Turing machine. Every Chaitin constant is simultaneously computably enumerable (the limit of a computable, increasing, converging sequence of rationals), and algorithmically random (its binary expansion is an algorithmic random sequence), hence uncomputable (Chaitin 1975).A Chaitin's constant can therefore be defined as(1)which gives the probability that for any set of instructions, a particular prefix-free universal Turing machine will halt, where is the size in bits of program .The value of a Chaitin constant is highly machine-dependent. In some cases, it can even be proved that not a single bit can be computed (Solovay 2000).Chaitin constants are perhaps the most obvious specific example of uncomputable numbers. They are also known to be transcendental.Calude et..
The problem of finding the mean triangle area of a triangle with vertices picked inside a triangle with unit area was proposed by Watson (1865) and solved by Sylvester. It solution is a special case of the general formula for polygon triangle picking.Since the problem is affine, it can be solved by considering for simplicity an isosceles right triangle with unit leg lengths. Integrating the formula for the area of a triangle over the six coordinates of the vertices (and normalizing to the area of the triangle and region of integration by dividing by the integral of unity over the region) gives(1)(2)where(3)is the triangle area of a triangle with vertices , , and .The integral can be solved using computer algebra by breaking up the integration region using cylindrical algebraic decomposition. This results in 62 regions, 30 of which have distinct integrals, each of which can be directly integrated. Combining the results then gives the result(4)(Pfiefer..
The (not necessarily regular) tetrahedron of least volume circumscribed around a convex body with volume is not known. If is a parallelepiped, then the smallest-volume tetrahedron containing it has volume 9/2. It is conjectured that this is the worst possible fit for the general problem, but this remains unproved.
Pick three points , , and distributed independently and uniformly in a unit disk (i.e., in the interior of the unit circle). Then the average area of the triangle determined by these points is(1)Using disk point picking, this can be writtenas(2)where(3)A trigonometric substitution can then be used to remove the trigonometric functions and split the integral into(4)where(5)(6)However, the easiest way to evaluate the integral is using Crofton's formula and polar coordinates to yield a mean triangle area(7)for unit-radius disks (OEIS A189511), or(8)for unit-area disks (OEIS A093587; Woolhouse 1867; Solomon 1978; Pfiefer 1989; Zinani 2003). This problem is very closely related to Sylvester's four-point problem, and can be derived as the limit as of the general polygon triangle picking problem.The distribution of areas, illustrated above, is apparently not known exactly.The probability that three random points in a disk form an acute..
The mean triangle area of a triangle picked at random inside a unit cube is , with variance .The distribution of areas, illustrated above, is apparently not known exactly.The probability that a random triangle in a cube is obtuse is approximately .
Sphere tetrahedron picking is the selection of quadruples of of points corresponding to vertices of a tetrahedron with vertices on the surface of a sphere. random tetrahedra can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere, n, 4].Pick four points on a sphere. What is the probability that the tetrahedron having these points as polyhedron vertices contains the center of the sphere? In the one-dimensional case, the probability that a second point is on the opposite side of 1/2 is 1/2. In the two-dimensional case, pick two points. In order for the third to form a triangle containing the center, it must lie in the quadrant bisected by a line segment passing through the center of the circle and the bisector of the two points. This happens for one quadrant, so the probability is 1/4. Similarly, for a sphere the probability is one octant, or 1/8.Pick four points at random on the surface of a unit sphereusing(1)(2)(3)with..
Given a simplex of unit content in Euclidean -space, pick points uniformly and independently at random, and denote the expected content of their convex hull by . Exact values are known only for and 2.(1)(2)(Buchta 1984, 1986), giving the first few values 0, 1/3, 1/2, 3/5, 2/3, 5/7, ...(OEIS A026741 and A026741).(3)(4)where is a harmonic number (Buchta 1984, 1986), giving the first few values 0, 0, 1/12, 1/6, 43/180, 3/10, 197/560, 499/1260, ... (OEIS A093762 and A093763).Not much is known about , although(5)(Buchta 1983, 1986) and(6)(Buchta 1986).Furthermore, Buchta and Reitzner (2001) give an explicit formula for the expected volume of the convex hull of points chosen at random in a three-dimensional simplex for arbitrary .
Let two points and be picked randomly from a unit -dimensional hypercube. The expected distance between the points , i.e., the mean line segment length, is then(1)This multiple integral has been evaluated analytically only for small values of . The case corresponds to the line line picking between two random points in the interval .The first few values for are given in the following table.OEIS1--0.3333333333...2A0915050.5214054331...3A0730120.6617071822...4A1039830.7776656535...5A1039840.8785309152...6A1039850.9689420830...7A1039861.0515838734...8A1039871.1281653402...The function satisfies(2)(Anderssen et al. 1976), plotted above together with the actual values.M. Trott (pers. comm., Feb. 23, 2005) has devised an ingenious algorithm for reducing the -dimensional integral to an integral over a 1-dimensional integrand such that(3)The first few values are(4)(5)(6)(7)In the limit as , these..
Ball triangle picking is the selection of triples of points (corresponding to vertices of a general triangle) randomly placed inside a ball. random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball, n, 3].The distribution of areas of a triangle with vertices picked at random in a unit ball is illustrated above. The mean triangle area is(1)(Buchta and Müller 1984, Finch 2010). random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball, n, 3].The determination of the probability for obtaining an acute triangle by picking three points at random in the unit disk was generalized by Hall (1982) to the -dimensional ball. Buchta (1986) subsequently gave closed form evaluations for Hall's integrals. Let be the probability that three points chosen independently and uniformly from the -ball form an acute triangle, then (2)(3)These can be combined..
Let be the number of (0,1)-matrices with no adjacent 1s (in either columns or rows). For , 2, ..., is given by 2, 7, 63, 1234, ... (OEIS A006506).The hard square entropy constant is defined by(OEIS A085850). It is not known if this constanthas an exact representation.The quantity arises in statistical physics (Baxter et al. 1980, Pearce and Seaton 1988), and is known as the entropy per site of hard squares. A related constant known as the hard hexagon entropy constant can also be defined.
The number of ways a set of elements can be partitioned into nonempty subsets is called a Bell number and is denoted (not to be confused with the Bernoulli number, which is also commonly denoted ).For example, there are five ways the numbers can be partitioned: , , , , and , so ., and the first few Bell numbers for , 2, ... are 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, ... (OEIS A000110). The numbers of digits in for , 1, ... are given by 1, 6, 116, 1928, 27665, ... (OEIS A113015).Bell numbers are implemented in the WolframLanguage as BellB[n].Though Bell numbers have traditionally been attributed to E. T. Bell as a result of the general theory he developed in his 1934 paper (Bell 1934), the first systematic study of Bell numbers was made by Ramanujan in chapter 3 of his second notebook approximately 25-30 years prior to Bell's work (B. C. Berndt, pers. comm., Jan. 4 and 13, 2010).The first few prime Bell numbers occur at indices..
Three types of matrices can be obtained by writing Pascal's triangle as a lower triangular matrix and truncating appropriately: a symmetric matrix with , a lower triangular matrix with , and an upper triangular matrix with , where , 1, ..., . For example, for , these would be given by(1)(2)(3)The Pascal -matrix or order is implemented in the Wolfram Language as LinearAlgebra`PascalMatrix[n].These matrices have some amazing properties. In particular, their determinants are all equal to 1(4)and(5)(Edelman and Strang).Edelman and Strang give four proofs of the identity (5), themost straightforward of which is(6)(7)(8)(9)where Einstein summation has been used.
The central binomial coefficient is never squarefree for . This was proved true for all sufficiently large by Sárkőzy's theorem. Goetgheluck (1988) proved the conjecture true for and Vardi (1991) for . The conjecture was proved true in its entirety by Granville and Ramare (1996).
The trinomial triangle is a number triangle of trinomial coefficients. It can be obtained by starting with a row containing a single "1" and the next row containing three 1s and then letting subsequent row elements be computed by summing the elements above to the left, directly above, and above to the right:(OEIS A027907).The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened trinomial triangle.
A partial solution to the Erdős squarefree conjecture which states that the binomial coefficient is never squarefree for all sufficiently large . Sárkőzy (1985) showed that if is the square part of the binomial coefficient , thenwhere is the Riemann zeta function. An upper bound on of has been obtained.
Let be the probability that a random walk on a -D lattice returns to the origin. In 1921, Pólya proved that(1)but(2)for . Watson (1939), McCrea and Whipple (1940), Domb (1954), and Glasser and Zucker (1977) showed that(3)(OEIS A086230), where(4)(5)(6)(7)(8)(9)(OEIS A086231; Borwein and Bailey 2003, Ch. 2, Ex. 20) is the third of Watson's triple integrals modulo a multiplicative constant, is a complete elliptic integral of the first kind, is a Jacobi theta function, and is the gamma function.Closed forms for are not known, but Montroll (1956) showed that for ,(10)where(11)(12)and is a modified Bessel function of the first kind.Numerical values of from Montroll (1956) and Flajolet (Finch 2003) are given in the following table.OEIS3A0862300.3405374A0862320.1932065A0862330.1351786A0862340.1047157A0862350.08584498A0862360.0729126..
RSA numbers are difficult to-factor composite numbers having exactly two prime factors (i.e., so-called semiprimes) that were listed in the Factoring Challenge of RSA Security®--a challenge that is now withdrawn and no longer active.While RSA numbers are much smaller than the largest known primes, their factorization is significant because of the curious property of numbers that proving or disproving a number to be prime ("primality testing") seems to be much easier than actually identifying the factors of a number ("prime factorization"). Thus, while it is trivial to multiply two large numbers and together, it can be extremely difficult to determine the factors if only their product is given. With some ingenuity, this property can be used to create practical and efficient encryption systems for electronic data.RSA Laboratories sponsored the RSA Factoring Challenge to encourage research into computational..
A set in a first-countable space is dense in if , where is the set of limit points of . For example, the rational numbers are dense in the reals. In general, a subset of is dense if its set closure .A real number is said to be -dense iff, in the base- expansion of , every possible finite string of consecutive digits appears. If is -normal, then is also -dense. If, for some , is -dense, then is irrational. Finally, is -dense iff the sequence is dense (Bailey and Crandall 2001, 2003).
The positive integers 216 and appear in an obscure passage in Plato's The Republic. In this passage, Plato alludes to the fact that 216 is equal to , where 6 is one of the numbers representing marriage since it is the product to the female 2 and the male 3. Plato was also aware of the fact the sum of the cubes of the 3-4-5 Pythagorean triple is equal to (Livio 2002, p. 66).In Laws, Plato suggests that is the optimal number of citizens in a state because 1. It is the product of 12, 20, and 21. 2. The 12th part of it can still be divided by 12. 3. It has 59 proper divisors, including all numbers for 1 to 12 except 11, and 5038--which is very close to 5040--is divisible by 11 (Livio 2002, p. 65).
A surface of revolution is a surface generated by rotating a two-dimensional curve about an axis. The resulting surface therefore always has azimuthal symmetry. Examples of surfaces of revolution include the apple surface, cone (excluding the base), conical frustum (excluding the ends), cylinder (excluding the ends), Darwin-de Sitter spheroid, Gabriel's horn, hyperboloid, lemon surface, oblate spheroid, paraboloid, prolate spheroid, pseudosphere, sphere, spheroid, and torus (and its generalization, the toroid).The area element of the surface of revolution obtained by rotating the curve from to about the x-axis is(1)(2)so the surface area is(3)(4)(Apostol 1969, p. 286; Kaplan 1992, p. 251; Anton 1999, p. 380). If the curve is instead specified parametrically by , the surface area obtained by rotating the curve about the x-axis for if in this interval is given by(5)Similarly, the area of the surface of revolution..
Given four points chosen at random inside a unit cube, the average volume of the tetrahedron determined by these points is given by(1)where the polyhedron vertices are located at where , ..., 4, and the (signed) volume is given by the determinant(2)The integral is extremely difficult to compute, but the analytic result for the mean tetrahedron volume is(3)(OEIS A093524; Zinani 2003). Note that the result quoted in the reply to Seidov (2000) actually refers to the average volume for tetrahedron tetrahedron picking.
Consider the average length of a line segment determined by two points picked at random in the interior of an arbitrary triangle. This problem is not affine, so a simple formula in terms of the area or linear properties of the original triangle apparently does not exist.However, if the original triangle is chosen to be an isosceles right triangle with unit legs, then the average length of a line with endpoints chosen at random inside it is given by(1)(2)(3)(OEIS A093063; M. Trott, pers. comm., Mar. 10, 2004), which is numerically surprisingly close to .Similarly, if the original triangle is chosen to be an equilateral triangle with unit side lengths, then the average length of a line with endpoints chosen at random inside it is given by(4)(5)The integrand can be split up into the four pieces(6)(7)(8)(9)As illustrated above, symmetry immediately gives and , so(10)With some effort, the integrals and can be done analytically to give..
An obtuse triangle is a triangle in which one of the angles is an obtuse angle. (Obviously, only a single angle in a triangle can be obtuse or it wouldn't be a triangle.) A triangle must be either obtuse, acute, or right.From the law of cosines, for a triangle with side lengths , , and ,(1)with the angle opposite side . For an angle to be obtuse, . Therefore, an obtuse triangle satisfies one of , , or .An obtuse triangle can be dissected into no fewer than seven acutetriangles (Wells 1986, p. 71).A famous problem is to find the chance that three points picked randomly in a plane are the polygon vertices of an obtuse triangle (Eisenberg and Sullivan 1996). Unfortunately, the solution of the problem depends on the procedure used to pick the "random" points (Portnoy 1994). In fact, it is impossible to pick random variables which are uniformly distributed in the plane (Eisenberg and Sullivan 1996). Guy (1993) gives a variety of solutions to the..
Consider the distribution of distances between a point picked at random in the interior of a unit cube and on a face of the cube. The probability function, illustrated above, was found in (nearly) closed form by Mathai et al. (1999). After simplifying, correcting typos, and completing the integrals, gives the closed form(1)The first even raw moments for , 2, 4, ... are 1, 2/3, 11/18, 211/315, 187/225, 11798/10395, ....
Instead of picking two points from the interior of the cube, instead pick two points on different faces of the unit cube. In this case, the average distance between the points is(1)(OEIS A093066; Borwein and Bailey 2003, p. 26;Borwein et al. 2004, pp. 66-67). Interestingly,(2)as apparently first noted by M. Trott (pers. comm., Mar. 21, 2008).The two integrals above can be written in terms of sums as(3)(4)(Borwein et al. 2004, p. 67), where however appears to be classically divergent and perhaps must be interpreted in some regularized sense.Consider a line whose endpoints are picked at random on opposite sides of the unit cube. The probability density function for the length of this line is given by(5)(Mathai 1999; after simplification). The mean length is(6)(7)The first even raw moments for , 2, 4, ... are 1, 4/3, 167/90, 284/105, 931/225, 9868/1485, ....Consider a line whose endpoints are picked at random..
The average distance between two points chosen at random inside a unit cube (the case of hypercube line picking), sometimes known as the Robbins constant, is(1)(2)(3)(OEIS A073012; Robbins 1978, Le Lionnais 1983).The probability function as a function of line length, illustrated above, was found in (nearly) closed form by Mathai et al. (1999). After simplifying, correcting typos, and completing the integrals, gives the closed form(4)The first even raw moments for , 2, ... are 1, 1/2, 11/30, 211/630, 187/525, 3524083/6306300, ... (OEIS A160693 and A160694).Pick points on a cube, and space them as far apart as possible. The best value known for the minimum straight line distance between any two points is given in the following table. 51.118033988749861.0606601482100718190.86602540378463100.74999998333331110.70961617562351120.70710678118660130.70710678118660140.70710678118660150.625..
Square triangle picking is the selection of triples of points (corresponding to endpoints of a triangle) randomly placed inside a square. random triangles can be picked in a unit square in the Wolfram Language using the function RandomPoint[Rectangle, n, 3].Given three points chosen at random inside a unit square, the average area of the triangle determined by these points is given analytically by the multiple integrals(1)(2)Here, represent the polygon vertices of the triangle for , 2, 3, and the (signed) area of these triangles is given by the determinant(3)(4)The solution was first given by Woolhouse (1867). Since attempting to do the integrals by brute force result in intractable integrands, the best approach using computer algebra is to divide the six-dimensional region of integration into subregions using cylindrical algebraic decomposition such that the sign of does not change, do the integral in each region directly, and then..
Square line picking is the selection of pairs of points (corresponding to endpoints of a line segment) randomly placed inside a square. random line segments can be picked in a unit square in the Wolfram Language using the function RandomPoint[Rectangle, n, 2].Picking two points at random from the interior of a unit square, the average distance between them is the case of hypercube line picking, i.e.,(1)(2)(3)(OEIS A091505).The exact probability function is given by(4)(M. Trott, pers. comm., Mar. 11, 2004), and the corresponding distribution function by(5)From this, the mean distance can be computed, as can the variance of lengths,(6)(7)The statistical median is given by the rootof the quartic equation(8)which is approximately .The th raw moment is given for , 4, 6, ... as 1/3, 17/90, 29/210, 187/1575, 239/207, ... (OEIS A103304 and A103305).If, instead of picking two points from the interior of a square, two points are..
The mean triangle area of a triangle picked inside a regular hexagon with unit area is (Woolhouse 1867, Pfiefer 1989). This is a special case of a general polygon triangle picking result due to Alikoski (1939).The distribution of areas, illustrated above, is apparently not known exactly.
Sphere line picking is the selection of pairs of points corresponding to vertices of a line segment with endpoints on the surface of a sphere. random line segments can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere, n, 2].Pick two points at random on a unit sphere. The first one can be placed at the north pole, i.e., assigned the coordinate (0, 0, 1), without loss of generality. The second point is then chosen at random using sphere point picking, and so can be assigned coordinates(1)(2)(3)with and . The distance between first and second points is then(4)and solving for gives(5)Now the probability function for distance is then given by(6)(Solomon 1978, p. 163), since and . Here, .Therefore, somewhat surprisingly, large distances are the most common, contrary to most people's intuition. A plot of 15 random lines is shown above. The raw moments are(7)giving the first few as(8)(9)(10)(11)(OEIS..
Finch (2010) gives an overview of known results for random Gaussian triangles.Let the vertices of a triangle in dimensions be normal (normal) variates. The probability that a Gaussian triangle in dimensions is obtuse is(1)(2)(3)(4)(5)where is the gamma function, is the hypergeometric function, and is an incomplete beta function.For even ,(6)(Eisenberg and Sullivan 1996).The first few cases are explicitly(7)(8)(9)(10)(OEIS A102519 and A102520). The even cases are therefore 3/4, 15/32, 159/512, 867/4096, ... (OEIS A102556 and A102557) and the odd cases are , where , 9/8, 27/20, 837/560, ... (OEIS A102558 and A102559).
The Robbins constant is the mean line segment length, i.e., the expected distance between two points chosen at random in cube line picking, namely(1)(2)(3)(OEIS A073012; Robbins 1978, Le Lionnais 1983).
Ball tetrahedron picking is the selection of quadruples of points (corresponding to vertices of a general tetrahedron) randomly placed inside a ball. random tetrahedra can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball, n, 4].The mean tetrahedron volume of a tetrahedronformed by four random points in a unit ball is(OEIS A093591; Hostinsky 1925; Solomon 1978,p. 124; Zinani 2003).
Given an -ball of radius , find the distribution of the lengths of the lines determined by two points chosen at random within the ball. The probability distribution of lengths is given by(1)where(2)and(3)is a regularized beta function, with is an incomplete beta function and is a beta function (Tu and Fischbach 2000).The first few are(4)(5)(6)(7)The mean line segment lengths for and the first few dimensions are given by(8)(9)(10)(11)(OEIS A093530 and A093531 and OEIS A093532 and A093533), corresponding to line line picking, disk line picking, (3-D) ball line picking, and so on.
The mean triangle area of a triangle picked inside a regular -gon of unit area is(1)where (Alikoski 1939; Solomon 1978, p. 109; Croft et al. 1991, p. 54). Prior to Alikoski's work, only the special cases , 4, 6, 8, and had been determined. The first few cases are summarized in the following table, where is the largest root of(2)and is the largest root of(3)problem3triangle triangle picking4square triangle picking5pentagon triangle picking6hexagon triangle picking78910Amazingly, the algebraic degree of is equal to , where is the totient function, giving the first few terms for , 4, ... as 1, 1, 2, 1, 3, 2, 3, 2, 5, 2, 6, 3, 4, 4, 8, ... (OEIS A023022). Therefore, the only values of for which is rational are , 4, and 6.
The Eulerian number gives the number of permutations of having permutation ascents (Graham et al. 1994, p. 267). Note that a slightly different definition of Eulerian number is used by Comtet (1974), who defines the Eulerian number (sometimes also denoted ) as the number of permutation runs of length , and hence .The Eulerian numbers are given explicitly by the sum(1)(Comtet 1974, p. 243). The Eulerian numbers satisfy the sum identity(2)as well as Worpitzky's identity(3)Eulerian numbers also arise in the surprising context of integrating the sincfunction, and also in sums of the form(4)(5)where is the polylogarithm function. is therefore given by the coefficient of in(6) has the exponential generating function(7)The Eulerian numbers satisfy the recurrence relation(8)Special cases are given by(9)(10)(11)and summarized in the following table.OEIS, , , ...1A0002950, 1, 4, 11, 26, 57, 120, 247, 502, 1013, ...2A0004600,..
Arrange copies of the digits 1, ..., such that there is one digit between the 1s, two digits between the 2s, etc. For example, the unique (modulo reversal) solution is 231213, and the unique (again modulo reversal) solution is 23421314. Solutions to Langford's problem exist only if , so the next solutions occur for . There are 26 of these, as exhibited by Lloyd (1971). In lexicographically smallest order (i.e., small digits come first), the first few Langford sequences are 231213, 23421314, 14156742352637, 14167345236275, 15146735423627, ... (OEIS A050998).The number of solutions for , 4, 5, ... (modulo reversal of the digits) are 1, 1, 0, 0, 26, 150, 0, 0, 17792, 108144, ... (OEIS A014552). No formula is known for the number of solutions of a given order .
A prime circle of order is a free circular permutation of the numbers from 1 to with adjacent pairs summing to a prime. The number of prime circles for , 2, ..., are 1, 1, 1, 2, 48, 512, ... (OEIS A051252). The prime circles for the first few even orders are given in the table below.prime circles2468,
A prime partition of a positive integer is a set of primes which sum to . For example, there are three prime partitions of 7 sinceThe number of prime partitions of , 3, ... are 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, ... (OEIS A000607). If for prime and for composite, then the Euler transform gives the number of partitions of into prime parts (Sloane and Plouffe 1995, p. 21).The minimum number of primes needed to sum to , 3, ... are 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, ... (OEIS A051034). The maximum number of primes needed to sum to is just , 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, ... (OEIS A004526), corresponding to a representation in terms of all 2s for an even number or one 3 and the rest 2s for an odd number.The numbers which can be represented by a single prime are obviously the primes themselves. Composite numbers which can be represented as the sum of two primes are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, ... (OEIS A051035), and composite..
A number of the form , where is a positive rational number which is not the square of another rational number is called a pure quadratic surd. A number of the form , where is rational and is a pure quadratic surd is sometimes called a mixed quadratic surd (Hardy 1967, p. 20).Quadratic surds are sometimes also called quadratic irrationals.In 1770, Lagrange proved that any quadratic surd has a regular continued fraction which is periodic after some point. This result is known as Lagrange's continued fraction theorem.
The number of binary bits necessary to represent a number, given explicitly by(1)(2)where is the ceiling function, is the floor function, and is lg, the logarithm to base 2. For , 2, ..., the sequence of bit lengths is given by 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, ... (OEIS A036377). The function is given by the Wolfram Language function BitLength[n].
The number of different triangles which have integer side lengths and perimeter is(1)(2)(3)where is the partition function giving the number of ways of writing as a sum of exactly terms, is the nearest integer function, and is the floor function (Andrews 1979, Jordan et al. 1979, Honsberger 1985). A slightly complicated closed form is given by(4)The values of for , 2, ... are 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, ... (OEIS A005044), which is also Alcuin's sequence padded with two initial 0s.The generating function for is given by(5)(6)(7) also satisfies(8)It is not known if a triangle with integer sides, triangle medians, and area exists (although there are incorrect proofs of the impossibility in the literature). However, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz have shown that there are infinitely many triangles with rational sides (Heronian triangles) with two rational..
A sequencewhere is a Sheffer sequence, is invertible, and ranges over the real numbers. If is an associated Sheffer sequence, then is called a cross sequence. If , thenis called an Appell cross sequence.An example is the Laguerre polynomial.
In general, there is no unique matrix solution to the matrix equationEven in the case of parallel to , there are still multiple matrices that perform this transformation. For example, given , all the following matrices satisfy the above equation:Therefore, vector division cannot be uniquely defined in terms of matrices.However, if the vectors are represented by complex numbers or quaternions, vector division can be uniquely defined using the usual rules of complex division and quaternion algebra, respectively.
The weak law of large numbers (cf. the strong law of large numbers) is a result in probability theory also known as Bernoulli's theorem. Let , ..., be a sequence of independent and identically distributed random variables, each having a mean and standard deviation . Define a new variable(1)Then, as , the sample mean equals the population mean of each variable.(2)(3)(4)(5)In addition,(6)(7)(8)(9)Therefore, by the Chebyshev inequality, for all ,(10)As , it then follows that(11)(Khinchin 1929). Stated another way, the probability that the average for an arbitrary positive quantity approaches 1 as (Feller 1968, pp. 228-229).
Planck's's radiation function is the function(1)which is normalized so that(2)However, the function is sometimes also defined without the numerical normalization factor of (e.g., Abramowitz and Stegun 1972, p. 999).The first and second raw moments are(3)(4)where is Apéry's constant, but higher order raw moments do not exist since the corresponding integrals do not converge.It has a maximum at (OEIS A133838), where(5)and inflection points at (OEIS A133839) and (OEIS A133840), where(6)
Let and be the perimeters of the circumscribed and inscribed -gon and and the perimeters of the circumscribed and inscribed -gon. Then(1)(2)The first follows from the fact that side lengths of the polygons on a circle of radius are(3)(4)so(5)(6)But(7)(8)Using the identity(9)then gives(10)The second follows from(11)Using the identity(12)gives(13)(14)(15)(16)Successive application gives the Archimedes algorithm, which can be used to provide successive approximations to pi ().
Successive application of Archimedes' recurrence formula gives the Archimedes algorithm, which can be used to provide successive approximations to (pi). The algorithm is also called the Borchardt-Pfaff algorithm. Archimedes obtained the first rigorous approximation of by circumscribing and inscribing -gons on a circle. From Archimedes' recurrence formula, the circumferences and of the circumscribed and inscribed polygons are(1)(2)where(3)For a hexagon, and(4)(5)where . The first iteration of Archimedes' recurrence formula then gives(6)(7)(8)Additional iterations do not have simple closed forms, but the numerical approximations for , 1, 2, 3, 4 (corresponding to 6-, 12-, 24-, 48-, and 96-gons) are(9)(10)(11)(12)(13)By taking (a 96-gon) and using strict inequalities to convert irrational bounds to rational bounds at each step, Archimedes obtained the slightly looser result(14)..
The prime spiral, also known as Ulam's spiral, is a plot in which the positive integers are arranged in a spiral (left figure), with primes indicated in some way along the spiral. In the right plot above, primes are indicated in red and composites are indicated in yellow.The plot above shows a larger part of the spiral in which the primes are shown as dots.Unexpected patterns of diagonal lines are apparent in such a plot, as illustrated in the above grid. This construction was first made by Polish-American mathematician Stanislaw Ulam (1909-1986) in 1963 while doodling during a boring talk at a scientific meeting. While drawing a grid of lines, he decided to number the intersections according to a spiral pattern, and then began circling the numbers in the spiral that were primes. Surprisingly, the circled primes appeared to fall along a number of diagonal straight lines or, in Ulam's slightly more formal prose, it "appears to exhibit a strongly..
The Kakeya needle problems asks for the plane figure of least area in which a line segment of width 1 can be freely rotated (where translation of the segment is also allowed). Surprisingly, there is no minimum area (Besicovitch 1928). Another iterative construction which tends to as small an area as desired is called a Perron tree (Falconer 1990, Wells 1991).When the figure is restricted to be convex, the smallest region is an equilateral triangle of unit height. Wells (1991) states that Kakeya discovered this, while Falconer (1990) attributes it to Pál.If convexity is replaced by the weaker assumption of simply-connectedness, then the area can still be arbitrarily small, but if the set is required to be star-shaped, then is a known lower bound (Cunningham 1965).The smallest simple convex domain in which one can put a segment of length 1 which will coincide with itself when rotated by has area(OEIS A093823; Le Lionnais 1983). ..
Select three points at random on the circumference of a unit circle and find the distribution of areas of the resulting triangles determined by these three points.The first point can be assigned coordinates without loss of generality. Call the central angles from the first point to the second and third and . The range of can be restricted to because of symmetry, but can range from . Then(1)so(2)(3)Therefore,(4)(5)(6)(7)But(8)(9)(10)(11)Write (10) as(12)then(13)and(14)From (12),(15)(16)(17)(18)(19)so(20)Also,(21)(22)(23)(24)so(25)Combining (◇) and (◇) gives the meantriangle area as(26)(OEIS A093582).The first few moments are(27)(28)(29)(30)(31)(32)(OEIS A093583 and A093584and OEIS A093585 and A093586).The variance is therefore given by(33)The probability that the interior of the triangle determined by the three points picked at random on the circumference of a circle contains the origin is 1/4...
Given a unit circle, pick two points at random on its circumference, forming a chord. Without loss of generality, the first point can be taken as , and the second by , with (by symmetry, the range can be limited to instead of ). The distance between the two points is then(1)The average distance is then given by(2)The probability density function is obtained from(3)The raw moments are then(4)(5)(6)giving the first few as(7)(8)(9)(10)(11)(OEIS A000984 and OEIS A093581 and A001803), where the numerators of the odd terms are 4 times OEIS A061549.The central moments are(12)(13)(14)giving the skewness and kurtosisexcess as(15)(16)Bertrand's problem asks for the probability that a chord drawn at random on a circle of radius has length .
Pick any two relatively prime integers and , then the circle of radius centered at is known as a Ford circle. No matter what and how many s and s are picked, none of the Ford circles intersect (and all are tangent to the x-axis). This can be seen by examining the squared distance between the centers of the circles with and ,(1)Let be the sum of the radii(2)then(3)But , so and the distance between circle centers is the sum of the circle radii, with equality (and therefore tangency) iff . Ford circles are related to the Farey sequence (Conway and Guy 1996).If , , and are three consecutive terms in a Farey sequence, then the circles and are tangent at(4)and the circles and intersect in(5)Moreover, lies on the circumference of the semicircle with diameter and lies on the circumference of the semicircle with diameter (Apostol 1997, p. 101)...
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.In the Season 4 episode "Masterpiece" (2008) of the CBS-TV crime drama "Criminal Minds," the agents of the FBI Behavioral Analysis Unit are confronted by a serial killer who uses the Fibonacci number sequence to determine the number of victims for each of his killing episodes. In this episode, character Dr. Reid also notices that locations of the killings lie on the graph of a golden spiral, and going to the center of the spiral allows Reid to determine the location of the killer's base of operations.
The term "square" can be used to mean either a square number (" is the square of ") or a geometric figure consisting of a convex quadrilateral with sides of equal length that are positioned at right angles to each other as illustrated above. In other words, a square is a regular polygon with four sides.When used as a symbol, denotes a square geometric figure with given vertices, while is sometimes used to denote a graph product (Clark and Suen 2000).A square is a special case of an isosceles trapezoid, kite, parallelogram, quadrilateral, rectangle, rhombus, and trapezoid.The diagonals of a square bisect one another and are perpendicular (illustrated in red in the figure above). In addition, they bisect each pair of opposite angles (illustrated in blue).The perimeter of a square with side length is(1)and the area is(2)The inradius , circumradius , and area can be computed directly from the formulas for a general regular polygon..
Wang's conjecture states that if a set of tiles can tile the plane, then they can always be arranged to do so periodically (Wang 1961). The conjecture was refuted when Berger (1966) showed that an aperiodic set of tiles existed. Berger used tiles, but the number has subsequently been greatly reduced. In fact, Culik (1996) has reduced the number of colored square tiles to 13.For purely square tiles, Culik's record still stands as of Feb. 2009. For non-square tiles, it is much more complicated due to the Penrose tiles (2 tiles), the Robertson tiling (6 tiles), and various Ammann tilings (2-5 tiles).
The mean tetrahedron volume of a tetrahedron with vertices chosen at random inside another tetrahedron of unit volume is given by(1)(2)(OEIS A093525; Buchta and Reitzner 1992; Mannion1994; Schneider 1997, p. 170; Buchta and Reitzner 2001; Zinani 2003).This provides a disproof of the conjecture that the solution to this problem is a rational number (1/57 had been suggested by Croft et al. 1991, p. 54), and renders obsolete Solomon's statement that "Explicit values for random points in non-spherical regions such as tetrahedrons, parallelepipeds, etc., have apparently not yet been successfully calculated" (Solomon 1978, p. 124).Furthermore, Buchta and Reitzner (2001) give an explicit formula for the expected volume of the convex hull of points chosen at random in a three-dimensional simplex for arbitrary ...
Landau's problems are the four "unattackable" problems mentioned by Landau in the 1912 Fifth Congress of Mathematicians in Cambridge, namely: 1. The Goldbach conjecture, 2. Twin prime conjecture, 3. Legendre's conjecture that for every there exists a prime between and (Hardy and Wright 1979, p. 415; Ribenboim 1996, pp. 397-398), and 4. The conjecture that there are infinitely many primes of the form (Euler 1760; Mirsky 1949; Hardy and Wright 1979, p. 19; Ribenboim 1996, pp. 206-208). The first few such primes are 2, 5, 17, 37, 101, 197, 257, 401, ... (OEIS A002496). Although it is not known if there always exists a prime between and , Chen (1975) has shown that a number which is either a prime or semiprime does always satisfy this inequality. Moreover, there is always a prime between and where (Iwaniec and Pintz 1984; Hardy and Wright 1979, p. 415). The smallest primes between and for , 2, ..., are 2, 5, 11,..
A prime magic square is a magic square consisting only of prime numbers (although the number 1 is sometimes allowed in such squares). The left square is the prime magic square (containing a 1) having the smallest possible magic constant, and was discovered by Dudeney in 1917 (Dudeney 1970; Gardner 1984, p. 86). The second square is the magic square consisting of primes only having the smallest possible magic constant (Madachy 1979, p. 95; attributed to R. Ondrejka). The third square is the prime magic square consisting of primes in arithmetic progression () having the smallest possible magic constant of 3117 (Madachy 1979, p. 95; attributed to R. Ondrejka). The prime magic square on the right was found by A. W. Johnson, Jr. (Dewdney 1988).According to a 1913 proof of J. N. Muncey (cited in Gardner 1984, pp. 86-87), the smallest magic square composed of consecutive odd primes including..
The Narayan number for , 2, ... and , ..., gives a solution to several counting problems in combinatorics. For example, gives the number of expressions with pairs of parentheses that are correctly matched and contain distinct nestings. It also gives the number Dyck paths of length with exactly peaks.A closed-form expression of is given bywhere is a binomial coefficient.Summing over gives the Catalan numberEnumerating as a number triangle is called the Narayana triangle.
Catalan's triangle is the number triangle(1)(OEIS A009766) with entries given by(2)for . Each element is equal to the one above plus the one to the left. The sum of each row is equal to the last element of the next row and also equal to the Catalan number . Furthermore, .The coefficients also give the number of nonnegative partial sums of 1s and s, denoted by Bailey (1996), who gave the alternate form(3)(4)for .
If divides the numerator of the Bernoulli number for , then is called an irregular pair. For , the irregular pairs of various forms are for , for , none for , and for .
The jinc function is defined as(1)where is a Bessel function of the first kind, and satisfies . The derivative of the jinc function is given by(2)The function is sometimes normalized by multiplying by a factor of 2 so that (Siegman 1986, p. 729).The first real inflection point of the function occurs when(3)namely 2.29991033... (OEIS A133920).The unique real fixed point occurs at 0.48541702373... (OEIS A133921).
The apodization function(1)Its full width at half maximum is .Its instrument function is(2)(3)where is a Bessel function of the first kind. This function has a maximum of . To investigate the instrument function, define the dimensionless parameter and rewrite the instrument function as(4)Finding the full width at half maximumthen amounts to solving(5)which gives , so for , the full width at half maximum is(6)The maximum negative sidelobe of times the peak, and maximum positive sidelobe of 0.356044 times the peak.
An apodization function(1)having instrument function(2)The peak of is . The full width at half maximum of can found by setting to obtain(3)and solving for , yielding(4)Therefore, with ,(5)The extrema are given by taking the derivative of , substituting , and setting equal to 0(6)Solving this numerically gives sidelobes at 0.715148 (), 1.22951 (0.256749), 1.73544 (), ....
An apodization function, also called the Hann function, frequently used to reduce leakage in discrete Fourier transforms. The illustrations above show the Hanning function, its instrument function, and a blowup of the instrument function sidelobes. It is named after the Austrian meteorologist Julius von Hann (Blackman and Tukey 1959, pp. 98-99). The Hanning function is given by(1)(2)Its full width at half maximum is .It has instrument function(3)(4)To investigate the instrument function, define the dimensionless parameter and rewrite the instrument function as(5)The half-maximum can then be seen to occur at(6)so for , the full width at half maximum is(7)To find the extrema, take the derivative(8)and equate to zero. The first two roots are and 10.7061..., corresponding to the first sidelobe minimum () and maximum (), respectively...
An apodization function chosen to minimize the height of the highest sidelobe (Hamming and Tukey 1949, Blackman and Tukey 1959). The Hamming function is given by(1)and its full width at half maximum is .The corresponding instrument function is(2)This apodization function is close to the one produced by the requirement that the instrument function goes to 0 at . The FWHM is , the peak is 1.08, and the peak negative and positive sidelobes (in units of the peak) are and 0.00734934, respectively.From the apodization function, a general symmetric apodization function can be written as a Fourier series(3)where the coefficients satisfy(4)The corresponding instrument function is(5)To obtain an apodization function with zero at , use(6)so(7)(8)(9)(10)(11)
Min Max Min Max Re Im In one dimension, the Gaussian function is the probabilitydensity function of the normal distribution,(1)sometimes also called the frequency curve. The full width at half maximum (FWHM) for a Gaussian is found by finding the half-maximum points . The constant scaling factor can be ignored, so we must solve(2)But occurs at , so(3)Solving,(4)(5)(6)(7)The full width at half maximum is thereforegiven by(8)In two dimensions, the circular Gaussian function is the distribution function for uncorrelated variates and having a bivariate normal distribution and equal standard deviation ,(9)The corresponding elliptical Gaussian function corresponding to is given by(10)The Gaussian function can also be used as an apodizationfunction(11)shown above with the corresponding instrumentfunction. The instrument function is(12)which has maximum(13)As , equation (12) reduces to(14)The hypergeometric function is also..
The apodization functionIts full width at half maximum is .Its instrument function iswhich has a maximum of and full width at half maximum .
The apodization functionIts full width at half maximum is .Its instrument function iswhere is a Bessel function of the first kind. This has a maximum of , and full width at half maximum of .
An apodization function given by(1)which has full width at half maximum of . This function is defined so that the coefficients are approximations in the general expansion(2)to(3)(4)(5)which produce zeros of at and .The corresponding instrument function is(6)where is the sinc function. It is full width at half maximum is .
The apodization function(1)which is a generalization of the one-argument triangle function. Its full width at half maximum is .It has instrument function(2)where is the sinc function. The peak of is , and the full width at half maximum is given by setting and numerically solving(3)for , yielding(4)Therefore, with ,(5)The function is always positive, so there are no negative sidelobes. The extrema are given by differentiating with respect to , defining , and setting equal to 0,(6)Solving this numerically gives minima of 0 at , 2, 3, ..., and sidelobes of 0.047190, 0.01648, 0.00834029, ... at , 2.45892, 3.47089, ....
The cosine function is one of the basic functions encountered in trigonometry (the others being the cosecant, cotangent, secant, sine, and tangent). Let be an angle measured counterclockwise from the x-axis along the arc of the unit circle. Then is the horizontal coordinate of the arc endpoint.The common schoolbook definition of the cosine of an angle in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle adjacent to the angle and the hypotenuse, i.e.,(1)A convenient mnemonic for remembering the definition of the sine, cosine, and tangent is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).As a result of its definition, the cosine function is periodic with period . By the Pythagorean theorem, also obeys the identity(2) Min Max Re Im The definition of the cosine function can be extended to..
Let be a positive integer, then is defined as the set of all matrices in the modular group Gamma with . is a subgroup of . For any prime , the setis a fundamental region of the subgroup , where and (Apostol 1997).
A number in which the first decimal digits of the fractional part sum to 666 is known as an evil number (Pegg and Lomont 2004).However, the term "evil" is also used to denote nonnegative integers that have an even number of 1s in their binary expansions, the first few of which are 0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, ... (OEIS A001969), illustrated above as a binary plot. Numbers that are not evil are then known as odious numbers.Returning to Pegg's definition of evil, the fact that is evil was noted by Keith, while I. Honig (pers. comm., May 9, 2004) noted that the golden ratio is also evil. The following table gives a list of some common evil numbers (Pegg and Lomont 2004).Ramanujan constant 132hard hexagon entropy constant 137139140Stieltjes constant 142pi 144golden ratio 146146151Glaisher-Kinkelin constant 153cube line picking average length155Delian constant 156The probability of the digits of a given real number summing..
An almost integer is a number that is very close to an integer.Surprising examples are given by(1)which equals to within 5 digits and(2)which equals to within 16 digits (M. Trott, pers. comm., Dec. 7, 2004). The first of these comes from the half-angle formula identity(3)where 22 is the numerator of the convergent 22/7 to , so . It therefore follows that any pi approximation gives a near-identity of the form .Another surprising example involving both e andpi is(4)which can also be written as(5)(6)Here, is Gelfond's constant. Applying cosine a few more times gives(7)This curious near-identity was apparently noticed almost simultaneously around 1988 by N. J. A. Sloane, J. H. Conway, and S. Plouffe, but no satisfying explanation as to "why" is true has yet been discovered.Another nested cosine almost integer is given by(8)(P. Rolli, pers. comm., Feb. 19, 2004).An..
, sometimes also denoted (Abramowitz and Stegun 1972, p. 825; Comtet 1974, p. 94; Hardy and Wright 1979, p. 273; Conway and Guy 1996, p. 94; Andrews 1998, p. 1), gives the number of ways of writing the integer as a sum of positive integers, where the order of addends is not considered significant. By convention, partitions are usually ordered from largest to smallest (Skiena 1990, p. 51). For example, since 4 can be written(1)(2)(3)(4)(5)it follows that . is sometimes called the number of unrestricted partitions, and is implemented in the Wolfram Language as PartitionsP[n].The values of for , 2, ..., are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ... (OEIS A000041). The values of for , 1, ... are given by 1, 42, 190569292, 24061467864032622473692149727991, ... (OEIS A070177).The first few prime values of are 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (OEIS A049575), corresponding to indices 2, 3, 4, 5, 6, 13, 36, 77, 132,..
In a 1847 talk to the Académie des Sciences in Paris, Gabriel Lamé (1795-1870) claimed to have proven Fermat's last theorem. However, Joseph Liouville immediately pointed out an error in Lamé's result by pointing out that Lamé had incorrectly assumed unique factorization in the ring of -cyclotomic integers. Kummer had already studied the failure of unique factorization in cyclotomic fields and subsequently formulated a theory of ideals which was later further developed by Dedekind.Kummer was able to prove Fermat's last theorem for all prime exponents falling into a class he called "regular." "Irregular" primes are thus primes that are not a member of this class, and a prime is irregular iff divides the class number of the cyclotomic field generated by . Equivalently, but more conveniently, an odd prime is irregular iff divides the numerator of a Bernoulli number with .An infinite number..
A twin Pythagorean triple is a Pythagorean triple for which two values are consecutive integers. By definition, twin triplets are therefore primitive triples. Of the 16 primitive triples with hypotenuse less than 100, seven are twin triples. The first few twin triples, sorted by increasing , are (3, 4, 5), (5, 12, 13), (7, 24, 25), (20, 21, 29), (9, 40, 41), (11, 60, 61), (13, 84, 85), (15, 112, 113), ....The numbers of twin triples with hypotenuse less than 10, , , ... are 1, 7, 24, 74, ... (OEIS A101903).The first few leg-leg triplets are (3, 4, 5), (20, 21, 29), (119, 120, 169), (696, 697, 985), ... (OEIS A001652, A046090, and A001653). A closed form is available for the th such pair. Consider the general reduced solution , then the requirement that the legs be consecutive integers is(1)Rearranging gives(2)Defining(3)(4)then gives the Pell equation(5)Solutions to the Pell equation are given by(6)(7)so the lengths of the legs and and the hypotenuse..
A Heronian tetrahedron, also called a perfect tetrahedron, is a (not necessarily regular) tetrahedron whose sides, face areas, and volume are all rational numbers. It therefore is a tetrahedron all of whose faces are Heronian triangles and additionally that has rational volume. (Note that the volume of a tetrahedron can be computed using the Cayley-Menger determinant.)The integer Heronian tetrahedron having smallest maximum side length is the one with edge lengths 51, 52, 53, 80, 84, 117; faces (117, 80, 53), (117, 84, 51), (80, 84, 52), (53, 51, 52); face areas 1170, 1800, 1890, 2016; and volume 18144 (Buchholz 1992; Guy 1994, p. 191). This is the only integer Heronian triangle with all side lengths less than 157.The integer Heronian tetrahedron with smallest possible surface area and volume has edges 25, 39, 56, 120, 153, and 160; areas 420, 1404, 1872, and 2688 (for a total surface area of 6384); and volume 8064 (Buchholz 1992, Peterson..
A Pythagorean triple is a triple of positive integers , , and such that a right triangle exists with legs and hypotenuse . By the Pythagorean theorem, this is equivalent to finding positive integers , , and satisfying(1)The smallest and best-known Pythagorean triple is . The right triangle having these side lengths is sometimes called the 3, 4, 5 triangle.Plots of points in the -plane such that is a Pythagorean triple are shown above for successively larger bounds. These plots include negative values of and , and are therefore symmetric about both the x- and y-axes.Similarly, plots of points in the -plane such that is a Pythagorean triple are shown above for successively larger bounds.It is usual to consider only primitive Pythagorean triples (also called "reduced"triples) in which and are relatively prime, since other solutions can be generated trivially from the primitive ones. The primitive triples are illustrated above, and..
A Pythagorean triangle is a right triangle with integer side lengths (i.e., whose side lengths form a Pythagorean triple). A Pythagorean triangle with is known as a primitive right triangle.The inradius of a Pythagorean triangle is always a whole number sinceThe area of such a triangle is also a whole number since for primitive Pythagorean triples, one of or must be even, and for imprimitive triples, both and are even, sois always a positive integer.
Let there be integers with . The values represent the denominations of different coins, where these denominations have greatest common divisor of 1. The sums of money that can be represented using the given coins are then given by(1)where the are nonnegative integers giving the numbers of each coin used. If , it is obviously possibly to represent any quantity of money . However, in the general case, only some quantities can be produced. For example, if the allowed coins are , it is impossible to represent and 3, although all other quantities can be represented.Determining the function giving the greatest for which there is no solution is called the coin problem, or sometimes the money-changing problem. The largest such for a given problem is called the Frobenius number .The result(2)(3)(Nijenhuis and Wilf 1972) is mathematical folklore. The total number of such nonrepresentable amounts is given by(4)The largest nonrepresentable amounts for..
A perfect cuboid is a cuboid having integer side lengths,integer face diagonals(1)(2)(3)and an integer space diagonal(4)The problem of finding such a cuboid is also called the brick problem, diagonals problem, perfect box problem, perfect cuboid problem, or rational cuboid problem.No perfect cuboids are known despite an exhaustive search for all "odd sides" up to (Butler, pers. comm., Dec. 23, 2004).Solving the perfect cuboid problem is equivalent to solving the Diophantineequations(5)(6)(7)(8)A solution with integer space diagonal and two out of three face diagonals is , , and , giving , , , and , which was known to Euler. A solution giving integer space and face diagonals with only a single nonintegral polyhedron edge is , , and , giving , , , and .
An Euler brick is a cuboid that possesses integer edges and face diagonals(1)(2)(3)If the space diagonal is also an integer, the Euler brick is called a perfect cuboid, although no examples of perfect cuboids are currently known.The smallest Euler brick has sides and face polyhedron diagonals , , and , and was discovered by Halcke (1719; Dickson 2005, pp. 497-500). Kraitchik gave 257 cuboids with the odd edge less than 1 million (Guy 1994, p. 174). F. Helenius has compiled a list of the 5003 smallest (measured by the longest edge) Euler bricks. The first few are (240, 117, 44), (275, 252, 240), (693, 480, 140), (720, 132, 85), (792, 231, 160), ... (OEIS A031173, A031174, and A031175).Interest in this problem was high during the 18th century, and Saunderson (1740) found a parametric solution always giving Euler bricks (but not giving all possible Euler bricks), while in 1770 and 1772, Euler found at least two parametric solutions...
A harmonic series is a continued fraction-like series defined by(Havil 2003, p. 99).Examples are given in the following table.OEISharmonic expansionA054977[2, 1, 1, 1, 1, 1, 1, ...]A096622[0, 1, 0, 1, 4, 1, 4, ...]A075874[3, 0, 0, 3, 1, 5, 6, 5, ...]
A Pisot number is a positive algebraic integer greater than 1 all of whose conjugate elements have absolute value less than 1. A real quadratic algebraic integer greater than 1 and of degree 2 or 3 is a Pisot number if its norm is equal to . The golden ratio (denoted when considered as a Pisot number) is an example of a Pisot number since it has degree two and norm .The smallest Pisot number is given by the positive root (OEIS A060006) of(1)known as the plastic constant. This number was identified as the smallest known by Salem (1944), and proved to be the smallest possible by Siegel (1944).Pisot constants give rise to almost integers. For example, the larger the power to which is taken, the closer , where is the floor function, is to either 0 or 1 (Trott 2004). For example, the spectacular example is within of an integer (Trott 2004, pp. 8-9).The powers of for which this quantity is closer to 0 are 1, 3, 4, 5, 6, 7, 8, 11, 12, 14, 17, ... (OEIS A051016), while..
Let be a number field and let be an order in . Then the set of equivalence classes of invertible fractional ideals of forms a multiplicative Abelian group called the Picard group of .If is a maximal order, i.e., the ring of integers of , then every fractional ideal of is invertible and the Picard group of is the class group of . The order of the Picard group of is sometimes called the class number of . If is maximal, then the order of the Picard group is equal to the class number of .
Wallis's constant is the real solution (OEIS A007493) to the cubic equationIt was solved by Wallis to illustrate Newton's methodfor numerical equation solving.
Let be a prime number, thenwhere and are homogeneous polynomials in and with integer coefficients. Gauss (1965, p. 467) gives the coefficients of and up to .Kraitchik (1924) generalized Gauss's formula to odd squarefree integers . Then Gauss's formula can be written in the slightly simpler formwhere and have integer coefficients and are of degree and , respectively, with the totient function and a cyclotomic polynomial. In addition, is symmetric if is even; otherwise it is antisymmetric. is symmetric in most cases, but it antisymmetric if is of the form (Riesel 1994, p. 436). The following table gives the first few and s (Riesel 1994, pp. 436-442).51711
Let be the smallest dimension of a hypercube such that if the lines joining all pairs of corners are two-colored for any , a complete graph of one color with coplanar vertices will be forced. Stated colloquially, this definition is equivalent to considering every possible committee from some number of people and enumerating every pair of committees. Now assign each pair of committees to one of two groups, and find the smallest that will guarantee that there are four committees in which all pairs fall in the same group and all the people belong to an even number of committees (Hoffman 1998, p. 54).An answer was proved to exist by Graham and Rothschild (1971), who also provided the best known upper bound, given by(1)where Graham's number is recursively defined by(2)and(3)Here, is the so-called Knuth up-arrow notation. is often cited as the largest number that has ever been put to practical use (Exoo 2003).In chained arrow notation, satisfies..
Min Max Min Max Re Im The hyperbolic cosine integral, often called the "Chi function" for short, is defined by(1)where is the Euler-Mascheroni constant. The function is given by the Wolfram Language command CoshIntegral[z].The Chi function has a unique real root at (OEIS A133746).The derivative of is(2)and the integral is(3)
The double factorial of a positive integer is a generalization of the usual factorial defined by(1)Note that , by definition (Arfken 1985, p. 547).The origin of the notation appears not to not be widely known and is not mentioned in Cajori (1993).For , 1, 2, ..., the first few values are 1, 1, 2, 3, 8, 15, 48, 105, 384, ... (OEIS A006882). The numbers of decimal digits in for , 1, ... are 1, 4, 80, 1285, 17831, 228289, 2782857, 32828532, ... (OEIS A114488).The double factorial is implemented in the WolframLanguage as n!! or Factorial2[n].The double factorial is a special case of the multifactorial.The double factorial can be expressed in terms of the gammafunction by(2)(Arfken 1985, p. 548).The double factorial can also be extended to negative odd integers using the definition(3)(4)for , 1, ... (Arfken 1985, p. 547). Min Max Re Im Similarly, the double factorial can be extended to complex arguments as(5)There are many identities..
A factorion is an integer which is equal to the sum of factorials of its digits. There are exactly four such numbers:(1)(2)(3)(4)(OEIS A014080; Gardner 1978, Madachy 1979, Pickover 1995). Obviously, the factorion of an -digit number cannot exceed .
The (complete) gamma function is defined to be an extension of the factorial to complex and real number arguments. It is related to the factorial by(1)a slightly unfortunate notation due to Legendre which is now universally used instead of Gauss's simpler (Gauss 1812; Edwards 2001, p. 8).It is analytic everywhere except at , , , ..., and the residue at is(2)There are no points at which .The gamma function is implemented in the WolframLanguage as Gamma[z].There are a number of notational conventions in common use for indication of a power of a gamma functions. While authors such as Watson (1939) use (i.e., using a trigonometric function-like convention), it is also common to write .The gamma function can be defined as a definite integral for (Euler's integral form)(3)(4)or(5)The complete gamma function can be generalized to the upper incomplete gamma function and lower incomplete gamma function . Min Max Re Im Plots of the real and imaginary..
Let be defined as the power series whose th term has a coefficient equal to the th prime ,(1)(2)The function has a zero at (OEIS A088751). Now let be defined by(3)(4)(5)(OEIS A030018).Then N. Backhouse conjectured that(6)(7)(OEIS A072508). This limit was subsequently shown to exist by P. Flajolet. Note that , which follows from the radius of convergence of the reciprocal power series.The continued fraction of Backhouse's constant is [1, 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, ...] (OEIS A074269), which is also the same as the continued fraction of except for a leading 0 in the latter.
Like the entire harmonic series, the harmonicseries(1)taken over all primes also diverges, as first shown by Euler in 1737 (Nagell 1951, p. 59; Hardy and Wright 1979, pp. 17 and 22; Wells 1986, p. 41; Havil 2003, pp. 28-31), although it does so very slowly. The sum exceeds 1, 2, 3, ... after 3, 59, 361139, ... (OEIS A046024) primes.Its asymptotic behavior is given by(2)where(3)(OEIS A077761) is the Mertens constant (Hardy and Wright 1979, p. 351; Hardy 1999, p. 50; Havil 2003, p. 64).
Let be an infinite Abelian semigroup with linear order such that is the unit element and implies for . Define a Möbius function on by andfor , 3, .... Further suppose that (the true Möbius function) for all . Then Braun's conjecture states thatfor all .
The goat problem (or bull-tethering problem) considers a fenced circular field of radius with a goat (or bull, or other animal) tied to a point on the interior or exterior of the fence by means of a tether of length , and asks for the solution to various problems concerning how much of the field can be grazed.Tieing a goat to a point on the interior of the fence with radius 1 using a chain of length , consider the length of chain that must be used in order to allow the goat to graze exactly one half the area of the field. The answer is obtained by using the equation for a circle-circle intersection(1)Taking gives(2)plotted above. Setting (i.e., half of ) leads to the equation(3)which cannot be solved exactly, but which has approximate solution(4)(OEIS A133731).Now instead consider tieing the goat to the exterior of the fence (or equivalently, to the exterior of a silo whose horizontal cross section is a circle) with radius . Assume that , so that the goat is not..
A sultan has granted a commoner a chance to marry one of his daughters. The commoner will be presented with the daughters one at a time and, when each daughter is presented, the commoner will be told the daughter's dowry (which is fixed in advance). Upon being presented with a daughter, the commoner must immediately decide whether to accept or reject her (he is not allowed to return to a previously rejected daughter). However, the sultan will allow the marriage to take place only if the commoner picks the daughter with the overall highest dowry. Then what is the commoner's best strategy, assuming he knows nothing about the distribution of dowries (Mosteller 1987)?Since the commoner knows nothing about the distribution of the dowries, the best strategy is to wait until a certain number of daughters have been presented, then pick the highest dowry thereafter (Havil 2003, p. 134). The exact number to skip is determined by the condition that the..
The number , where 666 is the beast number and denotes a factorial. The number has approximately decimal digits.The number of trailing zeros in the Leviathan number is(1)(2)(Pickover 1995).
Legion's number of the first kind is defined as(1)(2)where 666 is the beast number. It has 1881 decimaldigits.Legion's number of the second kind is defined as(3)(4)It has approximately digits, and ends with trailing zeros.
Approximations to Khinchin's constant include(1)(2)(3)(4)which are correct to 9, 7, 6, and 5 digits, respectively (M. Hudson, pers. comm., Nov. 20, 2004).
The tangent numbers, also called a zag number, andgiven by(1)where is a Bernoulli number, are numbers that can be defined either in terms of a generating function given as the Maclaurin series of or as the numbers of alternating permutations on , 3, 5, 7, ... symbols (where permutations that are the reverses of one another counted as equivalent). The first few for , 2, ... are 1, 2, 16, 272, 7936, ... (OEIS A000182).For example, the reversal-nonequivalent alternating permutations on and 3 numbers are , and , , respectively.The tangent numbers have the generating function(2)(3)(4)Shanks (1967) defines a generalization of the tangent numbers by(5)where is a Dirichlet L-series, giving the special case(6)The following table gives the first few values of for , 2, ....OEIS1A0001821, 2, 16, 272, 7936, ...2A0004641, 11, 361, 24611, ...3A0001912, 46, 3362, 515086, ...4A0003184, 128, 16384, 4456448, ...5A0003204, 272, 55744, 23750912, ...6A0004116,..
A -automatic set is a set of integers whose base- representations form a regular language, i.e., a language accepted by a finite automaton or state machine. If bases and are incompatible (do not have a common power) and if an -automatic set and -automatic set are both of density 0 over the integers, then it is believed that is finite. However, this problem has not been settled.Some automatic sets, such as the 2-automatic consisting of numbers whose binary representations contain at most two 1s: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, ... (OEIS A048645) have a simple arithmetic expression. However, this is not the case for general -automatic sets.
Odd values of are 1, 1, 3, 5, 27, 89, 165, 585, ... (OEIS A051044), and occur with ever decreasing frequency as becomes large (unlike , for which the fraction of odd values remains roughly 50%). This follows from the pentagonal number theorem which gives(1)(2)(3)(Gordon and Ono 1997), so is odd iff is of the form , i.e., 1, 5, 12, 22, 35, ... or 2, 7, 15, 26, 40, ....The values of for which is prime are 3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072, 320397, ... (OEIS A035359), with no others for (Weisstein, May 6, 2000). These values correspond to 2, 2, 3, 5, 89, 29927, 444793, 602644050950309, ... (OEIS A051005). It is not known if is infinitely often prime, but Gordon and Ono (1997) proved that it is "almost always" divisible by any given power of 2 (1997).Gordon and Hughes (1981) showed that(4)and(5)where is an integer depending only on ...
An integer is -balanced for a prime if, among all nonzero binomial coefficients for , ..., (mod ), there are equal numbers of quadratic residues and nonresidues (mod ). Let be the set of integers , , that are -balanced. Among all the primes , only those with , 3, and 11 have .The following table gives the -balanced integers for small primes (OEIS A093755).2357111317
To enumerate a set of objects satisfying some set of properties means to explicitly produce a listing of all such objects. The problem of determining or counting all such solutions is known as the enumeration problem.A generating functionis said to enumerate (Hardy 1999, p. 85).
The -ball, denoted , is the interior of a sphere , and sometimes also called the -disk. (Although physicists often use the term "sphere" to mean the solid ball, mathematicians definitely do not!)The ball of radius centered at point is implemented in the Wolfram Language as Ball[x, y, z, r].The equation for the surface area of the -dimensional unit hypersphere gives the recurrence relation(1)Using then gives the hypercontent of the -ball of radius as(2)(Sommerville 1958, p. 136; Apostol 1974, p. 430; Conway and Sloane 1993). Strangely enough, the content reaches a maximum and then decreases towards 0 as increases. The point of maximal content of a unit -ball satisfies(3)(4)(5)where is the digamma function, is the gamma function, is the Euler-Mascheroni constant, and is a harmonic number. This equation cannot be solved analytically for , but the numerical solution to(6)is (OEIS A074455) (Wells 1986, p. 67)...
The number (the cube root of 2) which is to be constructed in the cube duplication problem. This number is not a Euclidean number although it is an algebraic number of third degree.It has decimal digits 1.25992104989... (OEIS A002580).Its continued fraction is [1, 3, 1, 5, 1, 1,4, 1, 1, 8, 1, 14, 1, ...] (OEIS A002945).
What is the sofa of greatest area which can be moved around a right-angled hallway of unit width? Hammersley (Croft et al. 1994) showed that(1)(OEIS A086118). Gerver (1992) found a sofa with larger area and provided arguments indicating that it is either optimal or close to it. The boundary of Gerver's sofa is a complicated shape composed of 18 arcs. Its area can be given by defining the constants , , , and by solving(2)(3)(4)(5)This gives(6)(7)(8)(9)Now define(10)where(11)(12)(13)Finally, define the functions(14)(15)(16)The area of the optimal sofa is then given by(17)(18)(Finch 2003).
Bezdek and Kuperberg (1991) have constructed packings of identical ellipsoidsof densities arbitrarily close to(OEIS A093824), greater than the maximum density of (OEIS A093825) that is possible for sphere packing (Sloane 1998), as established by proof of the Kepler conjecture. Furthermore, J. Wills has modified the ellipsoid packing to yield an even higher density of (Bezdek and Kuperberg 1991).Donev et al. (2004) showed that a maximally random jammed state of M&Ms chocolate candies has a packing density of about 68%, or 4% greater than spheres. Furthermore, Donev et al. (2004) also showed by computer simulations other ellipsoid packings resulted in random packing densities approaching that of the densest sphere packings, i.e., filling nearly 74% of space.
Define the packing density of a packing of spheres to be the fraction of a volume filled by the spheres. In three dimensions, there are three periodic packings for identical spheres: cubic lattice, face-centered cubic lattice, and hexagonal lattice. It was hypothesized by Kepler in 1611 that close packing (cubic or hexagonal, which have equivalent packing densities) is the densest possible, and this assertion is known as the Kepler conjecture. The problem of finding the densest packing of spheres (not necessarily periodic) is therefore known as the Kepler problem, where(OEIS A093825; Steinhaus 1999, p. 202;Wells 1986, p. 29; Wells 1991, p. 237).In 1831, Gauss managed to prove that the face-centered cubic is the densest lattice packing in three dimensions (Conway and Sloane 1993, p. 9), but the general conjecture remained open for many decades.While the Kepler conjecture is intuitively obvious, the proof remained..
A circle packing is an arrangement of circles inside a given boundary such that no two overlap and some (or all) of them are mutually tangent. The generalization to spheres is called a sphere packing. Tessellations of regular polygons correspond to particular circle packings (Williams 1979, pp. 35-41). There is a well-developed theory of circle packing in the context of discrete conformal mapping (Stephenson).The densest packing of circles in the plane is the hexagonal lattice of the bee's honeycomb (right figure; Steinhaus 1999, p. 202), which has a packing density of(1)(OEIS A093766; Wells 1986, p. 30). Gauss proved that the hexagonal lattice is the densest plane lattice packing, and in 1940, L. Fejes Tóth proved that the hexagonal lattice is indeed the densest of all possible plane packings.Surprisingly, the circular disk is not the least economical region for packing the plane. The "worst"..
Find the plane lamina of least area which is capable of covering any plane figure of unit generalized diameter. A unit circle is too small, but a hexagon circumscribed on the unit circle is larger than necessary. Pál (1920) showed that the hexagon can be reduced by cutting off two isosceles triangles on the corners of the hexagon which are tangent to the hexagon's incircle (Wells 1991; left figure above). Sprague subsequently demonstrated that an additional small curvilinear region could be removed (Wells 1991; right figure above). These constructions give upper bounds.The hexagon having inradius (giving a diameter of 1) has side length(1)and the area of this hexagon is(2)(OEIS A010527).In the above figure, the sagitta is given by(3)(4)and the other distances by(5)(6)so the area of one of the equilateral triangles removed in Pál's reduction is(7)(8)(9)(10)so the area left after removing two of these triangles is(11)(12)(13)(OEIS..
The answer to the question "which fits better, a round peg in a square hole, or a square peg in a round hole?" can be interpreted as asking which is larger, the ratio of the area of a circle to its circumscribed square, or the area of the square to its circumscribed circle? In two dimensions, the ratios are and , respectively. Therefore, a round peg fits better into a square hole than a square peg fits into a round hole (Wells 1986, p. 74).However, this result is true only in dimensions , and for , the unit -hypercube fits more closely into the -hypersphere than vice versa (Singmaster 1964; Wells 1986, p. 74). This can be demonstrated by noting that the formulas for the content of the unit -ball, the content of its circumscribed hypercube, and the content of its inscribed hypercube are given by(1)(2)(3)The ratios in question are then(4)(5)(Singmaster 1964). The ratio of these ratios is the transcendental equation(6)illustrated..
In 1979, Conway and Norton discovered an unexpected intimate connection between the monster group and the j-function. The Fourier expansion of is given by(1)(OEIS A000521), where and is the half-period ratio, and the dimensions of the first few irreducible representations of are 1, 196883, 21296876, 842609326, ... (OEIS A001379).In November 1978, J. McKay noticed that the -coefficient 196884 is exactly one more than the smallest dimension of nontrivial representations of the (Conway and Norton 1979). In fact, it turns out that the Fourier coefficients of can be expressed as linear combinations of these dimensions with small coefficients as follows:(2)(3)(4)(5)Borcherds (1992) later proved this relationship, which became known as monstrous moonshine. Amazingly, there turn out to be yet more deep connections between the monster group and the j-function...
A modulo multiplication group is a finite group of residue classes prime to under multiplication mod . is Abelian of group order , where is the totient function.A modulo multiplication group can be visualized by constructing its cycle graph. Cycle graphs are illustrated above for some low-order modulo multiplication groups. Such graphs are constructed by drawing labeled nodes, one for each element of the residue class, and connecting cycles obtained by iterating . Each edge of such a graph is bidirected, but they are commonly drawn using undirected edges with double edges used to indicate cycles of length two (Shanks 1993, pp. 85 and 87-92).The following table gives the modulo multiplication groups of small orders, together with their isomorphisms with respect to cyclic groups .groupelements2121, 221, 341, 2, 3, 421, 561, 2, 3, 4, 5, 641, 3, 5, 761, 2, 4, 5, 7, 841, 3, 7, 9101, 2, 3, 4, 5, 6, 7, 8, 9, 1041, 5, 7, 11121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,..
Define a Bouniakowsky polynomial as an irreducible polynomial with integer coefficients, degree , and . The Bouniakowsky conjecture states that is prime for an infinite number of integers (Bouniakowsky 1857). As an example of the greatest common divisor caveat, the polynomial is irreducible, but always divisible by 2.Irreducible degree 1 polynomials () always generate an infinite number of primes by Dirichlet's theorem. The existence of a Bouniakowsky polynomial that can produce an infinitude of primes is undetermined. The weaker fifth Hardy-Littlewood conjecture asserts that is prime for an infinite number of integers .Various prime-generating polynomialsare known, but none of these always generates a prime (Legendre).Worse yet, it is unknown if a general Bouniakowsky polynomial will always produce at least 1 prime number. For example, produces no primes until , 764400, 933660, ... (OEIS A122131)...
The expected number of real zeros of a random polynomial of degree if the coefficients are independent and distributed normally is given by(1)(2)(Kac 1943, Edelman and Kostlan 1995). Another form of the equation is given by(3)(Kostlan 1993, Edelman and Kostlan 1995). The plots above show the integrand (left) and numerical values of (red curve in right plot) for small . The first few values are 1, 1.29702, 1.49276, 1.64049, 1.7596, 1.85955, ....As ,(4)where(5)(6)(OEIS A093601; top curve in right plot above).The initial term was derived by Kac (1943).
A sequence of polynomials , for , 1, 2, ..., where is exactly of degree for all .
The sum of the first odd numbers is a square number,A sort of converse also exists, namely the difference of the th and st square numbers is the th odd number, which follows from
The th cubic number is a sum of consecutive odd numbers, for example(1)(2)(3)(4)etc. This identity follows from(5)It also follows from this fact that(6)
A fractional ideal is a generalization of an ideal in a ring . Instead, a fractional ideal is contained in the number field , but has the property that there is an element such that(1)is an ideal in . In particular, every element in can be written as a fraction, with a fixed denominator.(2)Note that the multiplication of two fractional ideals is another fractional ideal.For example, in the field , the set(3)is a fractional ideal because(4)Note that , where(5)and so is an inverse to .Given any fractional ideal there is always a fractional ideal such that . Consequently, the fractional ideals form an Abelian group by multiplication. The principal ideals generate a subgroup , and the quotient group is called the ideal class group.
A Dedekind ring is a commutative ring in whichthe following hold. 1. It is a Noetherian ring and a integraldomain. 2. It is the set of algebraic integers in itsfield of fractions. 3. Every nonzero prime ideal is also a maximalideal. Of course, in any ring, maximal ideals are always prime. The main example of a Dedekind domain is the ring of algebraic integers in a number field, an extension field of the rational numbers. An important consequence of the above axioms is that every ideal can be written uniquely as a product of prime ideals. This compensates for the possible failure of unique factorization of elements into irreducibles.
The extension ring obtained from a commutative unit ring (other than the trivial ring) when allowing division by all non-zero divisors. The ring of fractions of an integral domain is always a field.The term "ring of fractions" is sometimes used to denote any localization of a ring. The ring of fractions in the above meaning is then referred to as the total ring of fractions, and coincides with the localization with respect to the set of all non-zero divisors.When defining addition and multiplication of fractions, all that is required of the denominators is that they be multiplicatively closed, i.e., if , then ,(1)(2)Given a multiplicatively closed set in a ring , the ring of fractions is all elements of the form with and . Of course, it is required that and that fractions of the form and be considered equivalent. With the above definitions of addition and multiplication, this set forms a ring.The original ring may not embed in this ring of..
The ordered pair , where is the number of real embeddings of the number field and is the number of complex-conjugate pairs of embeddings. The degree of the number field is .
Min Max Re Im The hyperbolic cotangent is defined as(1)The notation is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). It is implemented in the Wolfram Language as Coth[z].The hyperbolic cotangent satisfies the identity(2)where is the hyperbolic cosecant.It has a unique real fixed point where(3)at (OEIS A085984), which is related to the Laplace limit in the solution of Kepler's equation.The derivative is given by(4)where is the hyperbolic cosecant, and the indefinite integral by(5)where is a constant of integration.The Laurent series of is given by(6)(7)(OEIS A002431 and A036278), where is a Bernoulli number and is a Bernoulli polynomial. An asymptotic series about infinity on the real line is given by(8)
The nested radical constant is the constant defined by(1)(2)(OEIS A072449).No closed-form expression is known for this constant (Finch 2003, p. 8; S. Plouffe, pers. comm., Aug. 29, 2008).
Min Max Re Im The elliptic lambda function is a -modular function defined on the upper half-plane by(1)where is the half-period ratio, is the nome(2)and are Jacobi theta functions.The elliptic lambda function is essentially the same as the inverse nome, the difference being that elliptic lambda function is a function of the half-period ratio , while the inverse nome is a function of the nome , where is itself a function of .It is implemented as the Wolfram Languagefunction ModularLambda[tau].The elliptic lambda function satisfies the functional equations(3)(4) has the series expansion(5)(OEIS A115977), and has the series expansion(6)(OEIS A029845; Conway and Norton 1979; Borweinand Borwein 1987, p. 117). gives the value of the elliptic modulus for which the complementary and normal complete elliptic integrals of the first kind are related by(7)i.e., the elliptic integral singular value for . It can be computed from(8)where(9)and..
A Lambert series is a series of the form(1)for . Then(2)(3)where(4)The particular case is sometimes denoted(5)(6)(7)for (Borwein and Borwein 1987, pp. 91 and 95), where is a q-polygamma function. Special cases and related sums include(8)(9)(10)(11)(12)(Borwein and Borwein 1997, pp. 91-92), which arise in the reciprocalFibonacci and reciprocal Lucas constants.Some beautiful series of this type include(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)where is the Möbius function, is the totient function, is the number of divisors of , is the q-polygamma function, is the divisor function, is the number of representations of in the form where and are rational integers (Hardy and Wright 1979), is a Jacobi elliptic function (Bailey et al. 2006), is the Liouville function, and is the least significant bit of ...
The division of two complex numbers can be accomplished by multiplying the numerator and denominator by the complex conjugate of the denominator, for example, with and , is given by(1)(2)(3)(4)(5)where denotes the complex conjugate. In component notation with ,(6)
Two complex numbers and are added together componentwise,In component form,(Krantz 1999, p. 1).
A factorial prime is a prime number of the form , where is a factorial. is prime for , 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 26951, 34790, 94550, 103040, 147855, 208003, ... (OEIS A002982), the largest of which are summarized in the following table.digitsdiscoverer107,707Marchal, Carmody, and Kuosa (Caldwell; May 2002)142,891Marchal, Carmody, and Kuosa (Caldwell; May 2002)429,390D. Domanov/PrimeGrid (Oct. 4, 2010)471,794J. Winskill/PrimeGrid (Dec. 14, 2010)700,177PrimeGrid (Aug. 30, 2013)1,015,843S. Fukui (Jul. 25, 2016; https://primes.utm.edu/primes/page.php?id=121944) is prime for , 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, ... (OEIS A002981; Wells 1986, p. 70), the largest of which are summarized in the following table.digitsdiscoverer107,707K. Davis..
A double factorial prime is a prime number of the form , where is a double factorial. is prime for , 4, 6, 8, 16, 26, 64, 82, 90, 118, 194, 214, 728, ... (OEIS A007749), the largest of which are summarized in the following table.digitsdiscoverer169,435S. Fukai (Jun. 5, 2015)229,924S. Fukai (Jun. 5, 2015)344,538S. Fukai (Apr. 21, 2016) is prime for , 1, 2, 518, 33416, 37310, 52608, 123998, ... (OEIS A080778), the largest of which are summarized in the following table.digitsdiscoverer112,762H. Jamke (Jan. 3, 2008)288,864S. Fukai (Jun. 5, 2015)
There are (at least) three types of Euler transforms (or transformations). The first is a set of transformations of hypergeometric functions, called Euler's hypergeometric transformations.The second type of Euler transform is a technique for series convergence improvement which takes a convergent alternating series(1)into a series with more rapid convergence to the same value to(2)where the forward difference is defined by(3)(Abramowitz and Stegun 1972; Beeler et al. 1972). Euler's hypergeometric and convergence improvement transformations are related by the fact that when is taken in the second of Euler's hypergeometric transformations(4)where is a hypergeometric function, it gives Euler's convergence improvement transformation of the series (Abramowitz and Stegun 1972, p. 555).The third type of Euler transform is a relationship between certain types of integer sequences (Sloane and Plouffe 1995, pp. 20-21)...
The Gauss map is a function from an oriented surface in Euclidean space to the unit sphere in . It associates to every point on the surface its oriented unit normal vector. Since the tangent space at a point on is parallel to the tangent space at its image point on the sphere, the differential can be considered as a map of the tangent space at into itself. The determinant of this map is the Gaussian curvature, and negative one-half of the trace is the mean curvature.Another meaning of the Gauss map is the function(Trott 2004, p. 44), where is the floor function, plotted above on the real line and in the complex plane.The related function is plotted above, where is the fractional part.The plots above show blowups of the absolute values of these functions (a version of the left figure appears in Trott 2004, p. 44)...
The fraction of odd values of the partition function P(n) is roughly 50%, independent of , whereas odd values of occur with ever decreasing frequency as becomes large. Kolberg (1959) proved that there are infinitely many even and odd values of .Leibniz noted that is prime for , 3, 4, 5, 6, but not 7. In fact, values of for which is prime are 2, 3, 4, 5, 6, 13, 36, 77, 132, 157, 168, 186, ... (OEIS A046063), corresponding to 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (OEIS A049575). Numbers which cannot be written as a product of are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, ... (OEIS A046064), corresponding to numbers of nonisomorphic Abelian groups which are not possible for any group order.Ramanujan conjectured a number of amazing and unexpected congruences involving . In particular, he proved(1)using Ramanujan's identity (Darling 1919; Hardy and Wright 1979; Drost 1997; Hardy 1999, pp. 87-88; Hirschhorn 1999). Ramanujan (1919) also showed that(2)and..
Niven's theorem states that if and are both rational, then the sine takes values 0, , and .Particular cases include(1)(2)(3)
The word "base" in mathematics is used to refer to a particular mathematical object that is used as a building block. The most common uses are the related concepts of the number system whose digits are used to represent numbers and the number system in which logarithms are defined. It can also be used to refer to the bottom edge or surface of a geometric figure.A real number can be represented using any integer number as a base (sometimes also called a radix or scale). The choice of a base yields to a representation of numbers known as a number system. In base , the digits 0, 1, ..., are used (where, by convention, for bases larger than 10, the symbols A, B, C, ... are generally used as symbols representing the decimal numbers 10, 11, 12, ...).The digits of a number in base (for integer ) can be obtained in the Wolfram Language using IntegerDigits[x, b].Let the base representation of a number be written(1)(e.g., ). Then, for example, the number 10 is..
The Feigenbaum constant is a universal constant for functions approaching chaos via period doubling. It was discovered by Feigenbaum in 1975 (Feigenbaum 1979) while studying the fixed points of the iterated function(1)and characterizes the geometric approach of the bifurcation parameter to its limiting value as the parameter is increased for fixed . The plot above is made by iterating equation (1) with several hundred times for a series of discrete but closely spaced values of , discarding the first hundred or so points before the iteration has settled down to its fixed points, and then plotting the points remaining.A similar plot that more directly shows the cycle may be constructed by plotting as a function of . The plot above (Trott, pers. comm.) shows the resulting curves for , 2, and 4.Let be the point at which a period -cycle appears, and denote the converged value by . Assuming geometric convergence, the difference between this value and..
A fixed point is a point that does not change upon application of a map, system of differential equations, etc. In particular, a fixed point of a function is a point such that(1)The fixed point of a function starting from an initial value can be computed in the Wolfram Language using FixedPoint[f, x]. Similarly, to get a list of the values obtained by iterating the function until a fixed point is reached, the command FixedPointList[f, x] can be used.The following table lists the smallest positive fixed points for several simple functions.functionfixed pointOEIScosecant1.1141571408A133866cosine0.7390851332A003957cotangent0.8603335890A069855hyperbolic cosecant0.9320200293A133867hyperbolic cosine----hyperbolic cotangent1.1996786402A085984hyperbolic secant0.7650099545A069814hyperbolic sine0--hyperbolic tangent0--inverse cosecant1.1141571408A133866inverse cosine0.7390851332A003957inverse cotangent0.8603335890A069855inverse..
The Dottie number is the name given by Kaplan (2007) to the unique real root of (namely, the unique real fixed point of the cosine function), which is 0.739085... (OEIS A003957). The name "Dottie" is of no fundamental mathematical significance since it refers to a particular French professor who--no doubt like many other calculator uses before and after her--noticed that whenever she typed a number into her calculator and hit the cosine button repeatedly, the result always converged to this value.The number is well-known, having appeared in numerous elementary works on algebra already by the late 1880s (e.g., Bertrand 1865, p. 285; Heis 1886, p. 468; Briot 1881, pp. 341-343), and probably much earlier as well. It is also known simply as the cosine constant, cosine superposition constant, iterated cosine constant, or cosine fixed point constant. Arakelian (1981, pp. 135-136; 1995) has used the Armenian..
First published in Riemann's groundbreaking 1859 paper (Riemann 1859), the Riemann hypothesis is a deep mathematical conjecture which states that the nontrivial Riemann zeta function zeros, i.e., the values of other than , , , ... such that (where is the Riemann zeta function) all lie on the "critical line" (where denotes the real part of ).A more general statement known as the generalized Riemann hypothesis conjectures that neither the Riemann zeta function nor any Dirichlet L-series has a zero with real part larger than 1/2.Legend holds that the copy of Riemann's collected works found in Hurwitz's library after his death would automatically fall open to the page on which the Riemann hypothesis was stated (Edwards 2001, p. ix).While it was long believed that Riemann's hypothesis was the result of deep intuition on the part of Riemann, an examination of his papers by C. L. Siegel showed that Riemann had made detailed..
Let be the Riemann-Siegel function. The unique value such that(1)where , 1, ... is then known as a Gram point (Edwards 2001, pp. 125-126).An excellent approximation for Gram point can be obtained by using the first few terms in the asymptotic expansion for and inverting to obtain(2)where is the Lambert W-function. This approximation gives as error of for , decreasing to by .The following table gives the first few Gram points.OEIS0A11485717.84559954041A11485823.1702827012227.6701822178331.7179799547435.4671842971538.9992099640642.3635503920745.5930289815848.7107766217951.73384281331054.6752374468The integers closest to these points are 18, 23, 28, 32, 35, 39, 42, 46, 49, 52,55, 58, ... (OEIS A002505).There is a unique point at which , given by the solution to the equation(3)and having numerical value(4)(OEIS A114893).It is usually the case that . Values of for which this does not hold are , 134, 195, 211, 232, 254,..
The tangent function is defined by(1)where is the sine function and is the cosine function. The notation is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix).The common schoolbook definition of the tangent of an angle in a right triangle (which is equivalent to the definition just given) is as the ratio of the side lengths opposite to the angle and adjacent the angle, i.e.,(2)A convenient mnemonic for remembering the definition of the sine, cosine, and tangent is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).The word "tangent" also has an important related meaning as a line or plane which touches a given curve or solid at a single point. These geometrical objects are then called a tangent line or tangent plane, respectively. Min Max Re Im The definition of the tangent function can be extended to complex arguments using the definition(3)(4)(5)(6)where..
By analogy with the sinc function, define the tancfunction by(1)Since is not a cardinal function, the "analogy" with the sinc function is one of functional structure, not mathematical properties. It is quite possible that a better term than , as introduced here, could be coined, although there appears to be no name previously assigned to this function.The derivative is given by(2)The indefinite integral can apparently not be done in closed form in terms of conventionally defined functions.This function commonly arises in problems in physics, where it is desired to determine values of for which , i.e., . This is a transcendental equation whose first few solutions are given in the following table and illustrated above.OEISroot001A1153654.4934094579090641753...27.7252518369377071642...310.904121659428899827...414.066193912831473480...517.220755271930768739...The positive solutions can be written explicitly..
Replacing the logistic equation(1)with the quadratic recurrence equation(2)where (sometimes also denoted ) is a positive constant sometimes known as the "biotic potential" gives the so-called logistic map. This quadratic map is capable of very complicated behavior. While John von Neumann had suggested using the logistic map as a random number generator in the late 1940s, it was not until work by W. Ricker in 1954 and detailed analytic studies of logistic maps beginning in the 1950s with Paul Stein and Stanislaw Ulam that the complicated properties of this type of map beyond simple oscillatory behavior were widely noted (Wolfram 2002, pp. 918-919).The first few iterations of the logistic map (2) give(3)(4)(5)where is the initial value, plotted above through five iterations (with increasing iteration number indicated by colors; 1 is red, 2 is yellow, 3 is green, 4 is blue, and 5 is violet) for various values of .The..
By analogy with the tanc function, define the tanhcfunction by(1)It has derivative(2)The indefinite integral can apparently not be done in closed form in terms of conventionally defined functions.It has maximum at , and positive inflection point at the solution to(3)which is 0.919937667... (OEIS A133919).It has a unique real fixed point at 0.82242927726... (OEIS A133918).
The functions(1)(2)(3)(4) has an inflection point at(5)which can be solved numerically to give (OEIS A118080).
By analogy with the sinc function, define the sinhcfunction by(1)Since is not a cardinal function, the "analogy" with the sinc function is one of functional structure, not mathematical properties. It is quite possible that a better term than could be coined, although there appears to be no other name previously assigned to this function.The function has derivative(2)and indefinite integral(3)where is the Shi function.The function has real fixed points at 1.31328371835... (OEIS A133916)and 2.63924951389... (OEIS A133917).
The prime zeta function(1)where the sum is taken over primes is a generalizationof the Riemann zeta function(2)where the sum is over all positive integers. In other words, the prime zeta function is the Dirichlet generating function of the characteristic function of the primes . is illustrated above on positive the real axis, where the imaginary part is indicated in yellow and the real part in red. (The sign difference in the imaginary part compared to the plot appearing in Fröberg is presumably a result of the use of a different convention for .)Various terms and notations are used for this function. The term "prime zeta function" and notation were used by Fröberg (1968), whereas Cohen (2000) uses the notation .The series converges absolutely for , where , can be analytically continued to the strip (Fröberg 1968), but not beyond the line (Landau and Walfisz 1920, Fröberg 1968) due to the clustering of singular..
The Sierpiński sieve is a fractal described by Sierpiński in 1915 and appearing in Italian art from the 13th century (Wolfram 2002, p. 43). It is also called the Sierpiński gasket or Sierpiński triangle. The curve can be written as a Lindenmayer system with initial string "FXF--FF--FF", string rewriting rules "F" -> "FF", "X" -> "--FXF++FXF++FXF--", and angle .The th iteration of the Sierpiński sieve is implemented in the Wolfram Language as SierpinskiMesh[n].Let be the number of black triangles after iteration , the length of a side of a triangle, and the fractional area which is black after the th iteration. Then(1)(2)(3)The capacity dimension is therefore(4)(5)(6)(7)(OEIS A020857; Wolfram 1984; Borwein and Bailey2003, p. 46).The Sierpiński sieve is produced by the beautiful recurrenceequation(8)where denote bitwise..
Define the juggler sequence for a positive integer as the sequence of numbers produced by the iteration(1)where denotes the floor function. For example, the sequence produced starting with the number 77 is 77, 675, 17537, 2322378, 1523, 59436, 243, 3787, 233046, 482, 21, 96, 9, 27, 140, 11, 36, 6, 2, 1.Rather surprisingly, all integers appear to eventually reach 1, a conjecture that holds at least up to (E. W. Weisstein, Jan. 23, 2006). The numbers of steps needed to reach 1 for starting values of , 2, ... are 0, 1, 6, 2, 5, 2, 4, 2, 7, 7, 4, 7, 4, 7, 6, 3, 4, 3, 9, 3, ... (OEIS A007320), plotted above. The high-water marks for numbers of steps are 0, 1, 6, 7, 9, 11, 17, 19, 43, 73, 75, 80, 88, 96, 107, 131, ... (OEIS A095908), which occur for starting values of 1, 2, 3, 9, 19, 25, 37, 77, 163, 193, 1119, ... (OEIS A094679).The smallest integers requiring steps to reach 1 for , 2, ... are 1, 2, 4, 16, 7, 5, 3, 9, 33, 19, 81, 25, 353, ... (OEIS A094670)...
Let be a compact connected subset of -dimensional Euclidean space. Gross (1964) and Stadje (1981) proved that there is a unique real number such that for all , , ..., , there exists with(1)The magic constant of is defined by(2)where(3)These numbers are also called dispersion numbers and rendezvous values. For any , Gross (1964) and Stadje (1981) proved that(4)If is a subinterval of the line and is a circular disk in the plane, then(5)If is a circle, then(6)(OEIS A060294). An expression for the magic constant of an ellipse in terms of its semimajor and semiminor axes lengths is not known. Nikolas and Yost (1988) showed that for a Reuleaux triangle (7)Denote the maximum value of in -dimensional space by . Thenwhere is the gamma function (Nikolas and Yost 1988).An unrelated quantity characteristic of a given magicsquare is also known as a magic constant...
The integer sequence defined by the recurrencerelation(1)with the initial conditions . This is the same recurrence relation as for the Perrin sequence, but with different initial conditions.The recurrence relation can be solved explicitly,giving(2)where is the th root of(3)Another form of the solution is(4)where is the th root of(5)The first few terms are 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, ... (OEIS A000931).The first few prime Padovan numbers are 2, 2, 3, 5, 7, 37, 151, 3329, 23833, ... (OEIS A100891), corresponding to indices ,3, 4, 5, 7, 8, 14, 19, 30, 37, 84, 128, 469, 666, 1262, 1573, 2003, 2210, 2289, 4163, 5553, 6567, 8561, 11230, 18737, 35834, 44259, 536485, ... (OEIS A112882). The search for prime numerators has been completed up to by E. W. Weisstein (Apr. 10, 2011), and the following table summarizes the largest known values.decimal digitsdiscoverer53648565518E. W. Weisstein (May 16, 2009)72773488874E. W. Weisstein..
While the Catalan numbers are the number of p-good paths from to (0,0) which do not cross the diagonal line, the super Catalan numbers count the number of lattice paths with diagonal steps from to (0,0) which do not touch the diagonal line .The super Catalan numbers are given by the recurrencerelation(1)(Comtet 1974), with . (Note that the expression in Vardi (1991, p. 198) contains two errors.) A closed form expression in terms of Legendre polynomials for is(2)(3)(Vardi 1991, p. 199). The first few super Catalan numbers are 1, 1, 3, 11, 45, 197, ... (OEIS A001003). These are often called the "little" Schröder numbers. Multiplying by 2 gives the usual ("large") Schröder numbers 2, 6, 22, 90, ... (OEIS A006318).The first few prime super Catalan numbers have indices 3, 4, 6, 10, 216, ... (OEIS A092839), with no others less than (Weisstein, Mar. 7, 2004), corresponding to the numbers 3, 11, 197,..
Given the generating functions defined by(1)(2)(3)(OEIS A051028, A051029,and A051030), then(4)Hirschhorn (1995) showed that(5)(6)(7)where(8)(9)Hirschhorn (1996) showed that checking the first seven cases to 6 is sufficient to prove the result.
The Catalan numbers on nonnegative integers are a set of numbers that arise in tree enumeration problems of the type, "In how many ways can a regular -gon be divided into triangles if different orientations are counted separately?" (Euler's polygon division problem). The solution is the Catalan number (Pólya 1956; Dörrie 1965; Honsberger 1973; Borwein and Bailey 2003, pp. 21-22), as graphically illustrated above (Dickau).Catalan numbers are commonly denoted (Graham et al. 1994; Stanley 1999b, p. 219; Pemmaraju and Skiena 2003, p. 169; this work) or (Goulden and Jackson 1983, p. 111), and less commonly (van Lint and Wilson 1992, p. 136).Catalan numbers are implemented in the WolframLanguage as CatalanNumber[n].The first few Catalan numbers for , 2, ... are 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ... (OEIS A000108).Explicit formulas for include(1)(2)(3)(4)(5)(6)(7)where..
The integer sequence defined by the recurrence(1)with the initial conditions , , . This recurrence relation is the same as that for the Padovan sequence but with different initial conditions. The first few terms for , 1, ..., are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, ... (OEIS A001608).The above cartoon (Amend 2005) shows an unconventional sports application of the Perrin sequence (right panel). (The left two panels instead apply the Fibonacci numbers). is the solution of a third-order linear homogeneous recurrence equation having characteristic equation(2)Denoting the roots of this equation by , , and , with the unique real root, the solution is then(3)Here,(4)is the plastic constant , which is also given by the limit(5)The asymptotic behavior of is(6)The first few primes in this sequence are 2, 3, 2, 5, 5, 7, 17, 29, 277, 367, 853, ... (OEIS A074788), which occur for terms , 3, 4, 5, 6, 7, 10, 12, 20, 21, 24, 34, 38, 75, 122, 166, 236, 355, 356, 930, 1042,..
Apéry's numbers are defined by(1)(2)(3)where is a binomial coefficient. The first few for , 1, 2, ... are 1, 5, 73, 1445, 33001, 819005, ... (OEIS A005259).The first few prime Apéry numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092826), which have indices , 2, 12, 24, ... (OEIS A092825).The case of Schmidt's problem expresses these numbers in the form(4)(Strehl 1993, 1994; Koepf 1998, p. 55).They are also given by the recurrence equation(5)with and (Beukers 1987).There is also an associated set of numbers(6)(7)(Beukers 1987), where is a generalized hypergeometric function. The values for , 1, ... are 1, 3, 19, 147, 1251, 11253, 104959, ... (OEIS A005258). The first few prime -numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092827), which have indices , 2, 6, 8, ... (OEIS A092828), with no others for (Weisstein, Mar. 8, 2004).The..
The Pell numbers are the numbers obtained by the s in the Lucas sequence with and . They correspond to the Pell polynomial . Similarly, the Pell-Lucas numbers are the s in the Lucas sequence with and , and correspond to the Pell-Lucas polynomial .The Pell numbers and Pell-Lucas numbers are also equal to(1)(2)where is a Fibonacci polynomial.The Pell and Pell-Lucas numbers satisfy the recurrencerelation(3)with initial conditions and for the Pell numbers and for the Pell-Lucas numbers.The th Pell and Pell-Lucas numbers are explicitly given by the Binet-type formulas(4)(5)The th Pell and Pell-Lucas numbers are given by the binomial sums(6)(7)respectively.The Pell and Pell-Lucas numbers satisfy the identities(8)(9)(10)and(11)(12)For , 1, ..., the Pell numbers are 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, ... (OEIS A000129).For a Pell number to be prime, it is necessary that be prime. The indices of (probable) prime Pell numbers are 2, 3, 5,..
The Cantor set , sometimes also called the Cantor comb or no middle third set (Cullen 1968, pp. 78-81), is given by taking the interval (set ), removing the open middle third (), removing the middle third of each of the two remaining pieces (), and continuing this procedure ad infinitum. It is therefore the set of points in the interval whose ternary expansions do not contain 1, illustrated above.The th iteration of the Cantor is implemented in the Wolfram Language as CantorMesh[n].Iterating the process 1 -> 101, 0 -> 000 starting with 1 gives the sequence 1, 101, 101000101, 101000101000000000101000101, .... The sequence of binary bits thus produced is therefore 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ... (OEIS A088917) whose th term is amazingly given by (mod 3), where is a (central) Delannoy number and is a Legendre polynomial (E. W. Weisstein, Apr. 9, 2006). The recurrence plot for this..
The transformation(1)(2)where is the fractional part of and is the floor function, that takes a continued fraction to .
The conjecture that, for any triangle,(1)where , , and are the vertex angles of the triangle and is the Brocard angle. The Abi-Khuzam inequality states that(2)(Yff 1963, Le Lionnais 1983, Abi-Khuzam and Boghossian 1989), which can be used to prove the conjecture (Abi-Khuzam 1974).The maximum value of occurs when two angles are equal, so taking , and using , the maximum occurs at the maximum of(3)which occurs when(4)Solving numerically gives (OEIS A133844), corresponding to a maximum value of approximately 0.440053 (OEIS A133845).
Consider the inequalityfor integer , where is the divisor function and is the Euler-Mascheroni constant. This holds for 7, 11, 13, 14, 15, 17, 19, ... (OEIS A091901), and is false for 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, and 5040 (OEIS A067698).Robin's theorem states that the truth of the inequality for all is equivalent to the Riemann hypothesis (Robin 1984; Havil 2003, p. 207).
Extend Hilbert's inequality by letting and(1)so that(2)Levin (1937) and Stečkin (1949) showed that(3)and(4)Mitrinovic et al. (1991) indicate that this constant is the best possible.
Archimedes' axiom, also known as the continuity axiom or Archimedes' lemma, survives in the writings of Eudoxus (Boyer and Merzbach 1991), but the term was first coined by the Austrian mathematician Otto Stolz (1883). It states that, given two magnitudes having a ratio, one can find a multiple of either which will exceed the other. This principle was the basis for the method of exhaustion, which Archimedes invented to solve problems of area and volume.Symbolically, the axiom states thatiff the appropriate one of following conditions is satisfied for integers and : 1. If , then . 2. If , then . 3. If , then . Formally, Archimedes' axiom states that if and are two line segments, then there exist a finite number of points , , ..., on such thatand is between and (Itô 1986, p. 611). A geometry in which Archimedes' lemma does not hold is called a non-Archimedean Geometry...
Let be the smallest tour length for points in a -D hypercube. Then there exists a smallest constant such that for all optimal tours in the hypercube,(1)and a constant such that for almost all optimal tours in the hypercube,(2)These constants satisfy the inequalities(3)(4)(5)(6)(7)(8)(9)(Fejes Tóth 1940, Verblunsky 1951, Few 1955, Beardwood et al. 1959),where(10) is the gamma function, is an expression involving Struve functions and Bessel functions of the second kind,(11)(OEIS A086306; Karloff 1989), and(12)(OEIS A086307; Goddyn 1990).In the limit ,(13)(14)(15)and(16)where(17)and is the best sphere packing density in -D space (Goddyn 1990, Moran 1984, Kabatyanskii and Levenshtein 1978). Steele and Snyder (1989) proved that the limit exists.Now consider the constant(18)so(19)Nonrigorous numerical estimates give (Johnson et al. 1996) and (Percus and Martin 1996).A certain self-avoiding space-filling function..
Strang's strange figures are the figures produced by plotting a periodic function as a function of an integer argument for , 2, .... Unexpected patterns and periodicities result from near-commensurabilities of certain rational numbers with the period (Richert 1992). Strang figures are shown above for a number of common functions.
A binary plot of an integer sequence is a plot of the binary representations of successive terms where each term is represented as a column of bits with 1s colored black and 0s colored white. The columns are then placed side-by-side to yield an array of colored squares. Several examples are shown above for the positive integers , square numbers , Fibonacci numbers , and binomial coefficients .Binary plots can be extended to rational number sequences by placing the binary representations of numerators on top, and denominators on bottom, as illustrated above for the sequence .Similarly, by using other bases and coloring the base- digits differently, binary plots can be extended to n-ary plots.
The term "product" refers to the result of one or more multiplications. For example, the mathematical statement would be read " times equals ," where is the product.More generally, it is possible to take the product of many different kinds of mathematical objects, including those that are not numbers. For example, the product of two sets is given by the Cartesian product. In topology, the product of spaces can be defined by using the product topology. The product of two groups, vector spaces, or modules is given by the direct product. In category theory, the product of objects is given using the category product.The product symbol is defined by(1)Useful product identities include(2)(3)
Primorial primes are primes of the form , where is the primorial of . A coordinated search for such primes is being conducted on PrimeGrid. is prime for , 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, ... (OEIS A057704; Guy 1994, pp. 7-8; Caldwell 1995). These correspond to with , 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, ... (OEIS A006794). The largest known primorial primes as of Nov. 2015 are summarized in the following table (Caldwell).digitsdiscoverer6845Dec. 1992365851PrimeGrid (Dec. 20, 2010)476311PrimeGrid (Mar. 5, 2012) (also known as a Euclid number) is prime for , 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, ... (OEIS A014545; Guy 1994, Caldwell 1995, Mudge 1997). These correspond to with , 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547,..
Let be the th prime, then the primorial (which is the analog of the usual factorial for prime numbers) is defined by(1)The values of for , 2, ..., are 2, 6, 30, 210, 2310, 30030, 510510, ... (OEIS A002110).It is sometimes convenient to define the primorial for values other than just the primes, in which case it is taken to be given by the product of all primes less than or equal to , i.e.,(2)where is the prime counting function. For , 2, ..., the first few values of are 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, ... (OEIS A034386).The logarithm of is closely related to the Chebyshev function , and a trivial rearrangement of the limit(3)gives(4)(Ruiz 1997; Finch 2003, p. 14; Pruitt), where eis the usual base of the natural logarithm.
The elliptic logarithm is generalization of integrals ofthe formfor real, which can be expressed in terms of logarithmic and inverse trigonometric functions, tofor and real. This integral can be done analytically, but has a complicated form involving incomplete elliptic integrals of the first kind with complex parameters. The plots above show the special case .The elliptic logarithm is implemented in the Wolfram Language as EllipticLog[x, y, a, b], where is an unfortunate and superfluous parameter that must be set to either or and which multiplies the above integral by a factor of .The inverse of the elliptic logarithm is the ellipticexponential function.
The elliptic exponential function gives the value of in the elliptic logarithmfor and real such that .It is implemented in the Wolfram Language as EllipticExp[u, a, b], which returns together with the superfluous parameter which multiplies the above integral by a factor of .The top plot above shows (red), (violet), and (blue) for . The other plots show in the complex plane.The plots above show in the complex plane for .As can be seen from the plots, the elliptic exponential function is doublyperiodic in the complex plane.
A semicubical parabola is a curve of the form(1)(i.e., it is half a cubic, and hence has power ). It has parametric equations(2)(3)and the polar equation(4)The evolute of the parabola is a particular case of the semicubical parabola also called Neile's parabola or the cuspidal cubic. In Cartesian coordinates, it has equation(5)which can also be written(6)The Tschirnhausen cubic catacausticis also a semicubical parabola.The semicubical parabola is the curve along which a particle descending under gravity describes equal vertical spacings within equal times, making it an isochronous curve. It was discovered by William Neile in 1657 and was the first nontrivial algebraic curve to have its arc length computed. Wallis published the method in 1659, giving Neile the credit (MacTutor Archive). The problem of finding the curve having this property had been posed by Leibniz in 1687 and was also solved by Huygens (MacTutor Archive).The semicubical..
The remainder obtained when dividing a polynomial by another polynomial . The polynomial remainder is implemented in the Wolfram Language as PolynomialRemainder[p, q, x], and is related to the polynomial quotient byFor example, the polynomial remainder of and is , corresponding to polynomial quotient .
The quotient of two polynomials and , discarding any polynomial remainder. Polynomial quotients are implemented in the Wolfram Language as PolynomialQuotient[p, q, x], and are related to the polynomial remainder byFor example, the polynomial quotient of and is , leaving remainder .
Synthetic division is a shortcut method for dividing two polynomials which can be used in place of the standard long division algorithm. This method reduces the dividend and divisor polynomials into a set of numeric values. After these values are processed, the resulting set of numeric outputs is used to construct the polynomial quotient and the polynomial remainder.For an example of synthetic division, consider dividing by . First, if a power of is missing from either polynomial, a term with that power and a zero coefficient must be inserted into the correct position in the respective polynomial. In this case the term is missing from the dividend while the term is missing from the divisor; therefore, is added between the quintic and the cubic terms of the dividend while is added between the cubic and the linear terms of the divisor:(1)and(2)respectively.Next, all the variables and their exponents () are removed from the dividend, leaving instead..
In every residue class modulo , there is exactly one integer polynomial with coefficients and . This polynomial is called the normal polynomial modulo in the class (Nagell 1951, p. 94).
Ruffini's rule a shortcut method for dividing a polynomial by a linear factor of the form which can be used in place of the standard long division algorithm. This method reduces the polynomial and the linear factor into a set of numeric values. After these values are processed, the resulting set of numeric outputs is used to construct the polynomial quotient and the polynomial remainder.Note that Ruffini's rule is a special case of the more generalized notion of synthetic division in which the divisor polynomial is a monic linear polynomial. Confusingly, Ruffini's rule is sometimes referred to as synthetic division, thus leading to the common misconception that the scope of synthetic division is significantly smaller than that of the long division algorithm.For an example of Ruffini's rule, consider divided by . First, if a power of is missing from the dividend, a term with that power and a zero coefficient must be inserted into the correct position..
A field which is complete with respect to a discrete valuation is called a local field if its field of residue classes is finite. The Hasse principle is one of the chief applications of local field theory. A local field with field characteristic is isomorphic to the field of power series in one variable whose coefficients are in a finite field. A local field of characteristic zero is either the p-adic numbers, or power series in a complex variable.
A cyclotomic field is obtained by adjoining a primitive root of unity , say , to the rational numbers . Since is primitive, is also an th root of unity and contains all of the th roots of unity,(1)For example, when and , the cyclotomic field is a quadratic field(2)(3)(4)where the coefficients are contained in .The Galois group of a cyclotomic field over the rationals is the multiplicative group of , the ring of integers (mod ). Hence, a cyclotomic field is a Abelian extension. Not all cyclotomic fields have unique factorization, for instance, , where .
Given a set of primes, a field is called a class field if it is a maximal normal extension of the rationals which splits all of the primes in , and if is the maximal set of primes split by K. Here the set is defined up to the equivalence relation of allowing a finite number of exceptions.The basic example is the set of primes congruent to 1 (mod 4),The class field for is because every such prime is expressible as the sum of two squares .
The Chebotarev density theorem is a complicated theorem in algebraic number theory which yields an asymptotic formula for the density of prime ideals of a number field that split in a certain way in an algebraic extension of . When the base field is the field of rational numbers, the theorem becomes much simpler.Let be a monic irreducible polynomial of degree with integer coefficients with root , let , let be the normal closure of , and let be a partition of , i.e., an ordered set of positive integers with . A prime is said to be unramified (over the number field ) if it does not divide the discriminant of . Let denote the set of unramified primes. Consider the set of unramified primes for which factors as modulo , where is irreducible modulo and has degree . Also define the density of primes in as follows:Now consider the Galois group of the number field . Since this is a subgroup of the symmetric group , every element of can be represented as a permutation of letters,..
The name for the set of integers modulo , denoted . If is a prime , then the modulus is a finite field .
A global field is either a number field, a function field on an algebraic curve, or an extension of transcendence degree one over a finite field. From a modern point of view, a global field may refer to a function field on a complex algebraic curve as well as one over a finite field. A global field contains a canonical subring, either the algebraic integers or the polynomials. By choosing a prime ideal in its subring, a global field can be topologically completed to give a local field. For example, the rational numbers are a global field. By choosing a prime number , the rationals can be completed in the p-adic norm to form the p-adic numbers .A global field is called global because of the special case of a complex algebraic curve, for which the field consists of global functions (i.e., functions that are defined everywhere). These functions differ from functions defined near a point, whose completion is called a local field. Under favorable conditions,..