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Petersson considered the absolutely converging DirichletL-series(1)Writing the denominator as(2)where(3)and(4)Petersson conjectured that and are always complex conjugate, which implies(5)and(6)This conjecture was proven by Deligne (1974), which also proved the tau conjecture as a special case. Deligne was awarded the Fields medal for his proof.

A perfect magic cube is a magic cube for which the rows, columns, pillars, space diagonals, and diagonals of each orthogonal slice sum to the same number (i.e., the magic constant ). While this terminology is standard in the published literature (Gardner 1976, Benson and Jacoby 1981, Gardner 1988, Pickover 2002), it has been suggested at various times that such cubes instead be termed Myers cubes, Myers diagonal cubes, or diagonal magic cube (Heinz).There is a trivial perfect magic cube of order one, but no perfect cubes exist for orders 2-4 (Schroeppel 1972; Benson and Jacoby 1981, pp. 23-25; Gardner 1988). While normal perfect magic cubes of orders 7 and 9 have been known since the late 1800s, it was long not known if perfect magic cubes of orders 5 or 6 could exist (Wells 1986, p. 72), although Schroeppel (1972) and Gardner (1988) note that any such cube must have a central value of 63. (Confusingly, Benson and Jacoby (1981, p. 5)..

On July 10, 2003, Eric Weisstein computed the numbers of (0,1)-matrices all of whose eigenvalues are real and positive, obtaining counts for , 2, ... of 1, 3, 25, 543, 29281, .... Based on agreement with OEIS A003024, Weisstein then conjectured that is equal to the number of labeled acyclic digraphs on vertices.This result was subsequently proved by McKay et al. (2003, 2004).

The tau conjecture, also known as Ramanujan's hypothesis after its proposer, states thatwhere is the tau function. This was proven by Deligne (1974) in the course of proving the more general Petersson conjecture. Deligne was awarded the Fields medal for his proof.

There are many unsolved problems in mathematics. Severalfamous problems which have recently been solved include: 1. The Pólya conjecture (disproven byHaselgrove 1958, smallest counterexample found by Tanaka 1980). 2. The four-color theorem (Appel and Haken1977ab and Appel et al. 1977 using a computer-assisted proof). 3. The Bieberbach conjecture (L. deBranges 1985). 4. Tait's flyping conjecture (Menasco and Thistlethwaite in 1991) and the other two of Tait's knot conjectures (by various authors 1987). 5. Fermat's last theorem (Wiles 1995, Taylorand Wiles 1995). 6. The Kepler conjecture (Hales 2002). 7. The Taniyama-Shimura conjecture(Breuil et al. in 1999). 8. The honeycomb conjecture (Hales 1999).9. The Poincaré conjecture. 10. Catalan's conjecture. 11. The strong perfect graph theorem...

A double bubble is pair of bubbles which intersect and are separated by a membrane bounded by the intersection. The usual double bubble is illustrated in the left figure above. A more exotic configuration in which one bubble is torus-shaped and the other is shaped like a dumbbell is illustrated at right (illustrations courtesy of J. M. Sullivan).In the plane, the analog of the double bubble consists of three circular arcs meeting in two points. It has been proved that the configuration of arcs meeting at equal angles) has the minimum perimeter for enclosing two equal areas (Alfaro et al. 1993, Morgan 1995).It had been conjectured that two equal partial spheres sharing a boundary of a flat disk separate two volumes of air using a total surface area that is less than any other boundary. This equal-volume case was proved by Hass et al. (1995), who reduced the problem to a set of integrals which they carried out on an ordinary PC. Frank Morgan,..

The set of fixed points which do not move as a knot is transformed into itself is not a knot. The conjecture was proved in 1978 (Morgan and Bass 1984). According to Morgan and Bass (1984), the Smith conjecture stands in the first rank of mathematical problems when measured by the amount and depth of new mathematics required to solve it.The generalized Smith conjecture considers to be a piecewise linear -dimensional hypersphere in , and the -fold cyclic covering of branched along , and asks if is unknotted if is an (Hartley 1983). This conjecture is true for , and false for , with counterexamples in the latter case provided by Giffen (1966), Gordon (1974), and Sumners (1975).

Checkers is a two-player game with the most common variant played on an checkerboard with each player starts with twelve pieces of a fixed color on opposite sites of the board. The most common variant of checkers is so-called "pool checkers," also called "Spanish pool checkers," draughts or draught (in the United Kingdom and some other countries), American checkers, and straight checkers. Play proceeds alternately between players, where all pieces may initially only move and capture in a forward diagonal direction. The allowable direction of play is modified for a piece if it is "crowned" by reaching the other side of the board, after which it may move either forwards or backwards. An opponent's piece may be captured by jumping over it diagonally, and the game is won by capturing all the opponents pieces or leaving the opponent with no legal moves.The most widely available sets of checkers consist of black and..

The theorem, originally conjectured by Berge (1960, 1961), that a graph is perfect iff neither the graph nor its graph complement contains an odd graph cycle of length at least five as an induced subgraph became known as the strong perfect graph conjecture (Golumbic 1980; Skiena 1990, p. 221). The conjecture can be stated more simply as the assertion that a graph is perfect iff it contains no odd graph hole and no odd graph antihole. The proposition can be stated even more succinctly as "a graph is perfect iff it is a Berge graph."This conjecture was proved in May 2002 following a remarkable sequence of results by Chudnovsky, Robertson, Seymour, and Thomas (Cornuéjols 2002, MacKenzie 2002).

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 th central binomial coefficient is defined as(1)(2)where is a binomial coefficient, is a factorial, and is a double factorial.These numbers have the generating function(3)The first few values are 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, ... (OEIS A000984). The numbers of decimal digits in for , 1, ... are 1, 6, 59, 601, 6019, 60204, 602057, 6020597, ... (OEIS A114501). These digits converge to the digits in the decimal expansion of (OEIS A114493).The central binomial coefficients are never prime except for .A scaled form of the central binomial coefficient is known as a Catalannumber(4)Erdős and Graham (1975) conjectured that the central binomial coefficient is never squarefree for , and this is sometimes known as the Erdős squarefree conjecture. Sárkőzy's theorem (Sárkőzy 1985) provides a partial solution which states that the binomial coefficient is never squarefree for all sufficiently..

Based on a problem in particle physics, Dyson (1962abc) conjectured that the constantterm in the Laurent seriesis the multinomial coefficientThe theorem was proved by Wilson (1962) and independently by Gunson (1962). A definitive proof was subsequently published by Good (1970).

If the Gauss map of a complete minimal surface omits a neighborhood of the sphere, then the surface is a plane. This was proven by Osserman (1959). Xavier (1981) subsequently generalized the result as follows. If the Gauss map of a complete minimal surface omits points, then the surface is a plane.

The conjecture that the equations for a Robbins algebra, commutativity, associativity,and the Robbins axiomwhere denotes NOT and denotes OR, imply those for a Boolean algebra. The conjecture was finally proven using a computer (McCune 1997).

In Note M, Burnside (1955) states, "The contrast that these results shew between groups of odd and of even order suggests inevitably that simple groups of odd order do not exist." Of course, simple groups of prime order do exist, namely the groups for any prime . Therefore, Burnside conjectured that every finite simple group of non-prime order must have even order. The conjecture was proven true by Feit and Thompson (1963).

Gelfond's theorem, also called the Gelfond-Schneider theorem, states that is transcendental if 1. is algebraic and 2. is algebraic and irrational. This provides a partial solution to the seventh of Hilbert'sproblems. It was proved independently by Gelfond (1934ab) and Schneider (1934ab).This establishes the transcendence of Gelfond's constant (since ) and the Gelfond-Schneider constant .Gelfond's theorem is implied by Schanuel's conjecture(Chow 1999).

A semiprime which English economist and logician William Stanley Jevons incorrectly believed no one else would be able to factor. According to Jevons (1874, p. 123), "Can the reader say what two numbers multiplied together will produce the number 8616460799? I think it unlikely that anyone but myself will ever know."Actually, a modern computer can factor this number in a few milliseconds as the product of two five-digit numbers:Published factorizations include those by Lehmer (1903) and Golomb (1996).

If replacing each number by its square in a magic square produces another magic square, the square is said to be a bimagic square. Bimagic squares are also called doubly magic squares, and are 2-multimagic squares.Lucas (1891) and later Hendricks (1998) showed that a bimagic square of order 3 is impossible for any set of numbers except the trivial case of using the same number 9 times.The first known bimagic square, constructed by Pfeffermann (1891a; left figure), had order 8 with magic constant 260 for the base square and after squaring. Another order 8 bimagic square is shown at right.Benson and Jacoby (1976) stated their belief that no bimagic squares of order less than 8 exist, and this was subsequently proved by Boyer and Trump in 2002 (Boyer).Pfeffermann (1891b) also published the first 9th-order bimagic square. Only a part of the first Pfeffermann's bimagic squares of both order 8 and of order 9 were published, with their completion left as..

A braid with strands and components with positive crossings and negative crossings satisfieswhere is the unknotting number. While the second part of the inequality was already known to be true (Boileau and Weber, 1983, 1984) at the time the conjecture was proposed, the proof of the entire conjecture was completed using results of Kronheimer and Mrowka on Milnor's conjecture (and, independently, using the slice-Bennequin inequality).

The prime number theorem gives an asymptotic form for the prime counting function , which counts the number of primes less than some integer . Legendre (1808) suggested that for large ,(1)with (where is sometimes called Legendre's constant), a formula which is correct in the leading term only,(2)(Nagell 1951, p. 54; Wagon 1991, pp. 28-29; Havil 2003, p. 177).In 1792, when only 15 years old, Gauss proposed that(3)Gauss later refined his estimate to(4)where(5)is the logarithmic integral. Gauss did not publish this result, which he first mentioned in an 1849 letter to Encke. It was subsequently posthumously published in 1863 (Gauss 1863; Havil 2003, pp. 174-176).Note that has the asymptotic series about of(6)(7)and taking the first three terms has been shown to be a better estimate than alone (Derbyshire 2004, pp. 116-117).The statement (4) is often known as "the" prime number theorem and was proved..

Smale's problems are a list of 18 challenging problems for the twenty-first century proposed by Field medalist Steven Smale. These problems were inspired in part by Hilbert's famous list of problems presented in 1900 (Hilbert's problems), and in part in response to a suggestion by V. I. Arnold on behalf of the International Mathematical Union that mathematicians describe a number of outstanding problems for the 21st century.1. The Riemann hypothesis. 2. The Poincaré conjecture. 3. Does (i.e., are P-problems equivalent to NP-problems)? 4. Integer zeros of a polynomial. 5. Height bounds for Diophantine curves. 6. Finiteness of the number of relative equilibria in celestial mechanics. 7. Distribution of points on the 2-sphere. 8. Introduction of dynamics into economic theory. 9. The linear programming problem. 10. The closing lemma. 11. Is 1-dimensional dynamics generally hyperbolic? 12. Centralizers of diffeomorphisms...

Hilbert's problems are a set of (originally) unsolved problems in mathematics proposed by Hilbert. Of the 23 total appearing in the printed address, ten were actually presented at the Second International Congress in Paris on August 8, 1900. In particular, the problems presented by Hilbert were 1, 2, 6, 7, 8, 13, 16, 19, 21, and 22 (Derbyshire 2004, p. 377). Furthermore, the final list of 23 problems omitted one additional problem on proof theory (Thiele 2001).Hilbert's problems were designed to serve as examples for the kinds of problems whose solutions would lead to the furthering of disciplines in mathematics. As such, some were areas for investigation and therefore not strictly "problems."1. "Cantor's problem of the cardinal number of the continuum." The question of if there is a transfinite number between that of a denumerable set and the numbers of the continuum was answered by Gödel and Cohen in their..

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 bound for the number of colors which are sufficient for map coloring on a surface of genus ,is the best possible, where is the floor function. is called the chromatic number, and the first few values for , 1, ... are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, ... (OEIS A000934).The fact that is also necessary was proved by Ringel and Youngs (1968) with two exceptions: the sphere (and plane), and the Klein bottle. When the four-color theorem was proved in 1976, the Klein bottle was left as the only exception, in that the Heawood formula gives seven, but the correct bound is six (as demonstrated by the Franklin graph). The four most difficult cases to prove in the Heawood conjecture were , 83, 158, and 257.

If , , ... are sets of positive integers andthen some contains arbitrarily long arithmetic progressions. The conjecture was proved by van der Waerden (1927) and is now known as van der Waerden's Theorem.According to de Bruijn (1977), "We do not know when and in what context he [Baudet] stated his conjecture and what partial results he had," although van der Waerden (1971, 1998) indicates he first heard of the problem in 1926.

A combinatorial conjecture formulated by Kneser (1955). It states that whenever the -subsets of a -set are divided into classes, then two disjoint subsets end up in the same class.Lovász (1978) gave a proof based on graph theory. In particular, he showed that the Kneser graph, whose vertices represent the -subsets, and where each edge connects two disjoint subsets, is not -colorable. More precisely, his results says that the chromatic number is equal to , and this implies that Kneser's conjecture is always false if the number of classes is increased to .An alternate proof was given by Bárány (1978).

The numerators and denominators obtained by taking the ratios of adjacent terms in the triangular array of the number of "bordered" alternating sign matrices with a 1 at the top of column are, respectively, the numbers in the (2, 1)- and (1, 2)-Pascal triangles which are different from 1. This conjecture was proven by Zeilberger (1996).

The conjecture that the number of alternating sign matrices "bordered" by s is explicitly given by the formulaThis conjecture was proved by Doron Zeilberger in 1995 (Zeilberger 1996a). This proof enlisted the aid of an army of 88 referees together with extensive computer calculations. A beautiful, shorter proof was given later that year by Kuperberg (Kuperberg 1996), and the refined alternating sign matrix conjecture was subsequently proved by Zeilberger (Zeilberger 1996b) using Kuperberg's method together with techniques from -calculus and orthogonal polynomials.

Goldbach's original conjecture (sometimes called the "ternary" Goldbach conjecture), written in a June 7, 1742 letter to Euler, states "at least it seems that every number that is greater than 2 is the sum of three primes" (Goldbach 1742; Dickson 2005, p. 421). Note that Goldbach considered the number 1 to be a prime, a convention that is no longer followed. As re-expressed by Euler, an equivalent form of this conjecture (called the "strong" or "binary" Goldbach conjecture) asserts that all positive even integers can be expressed as the sum of two primes. Two primes such that for a positive integer are sometimes called a Goldbach partition (Oliveira e Silva).According to Hardy (1999, p. 19), "It is comparatively easy to make clever guesses; indeed there are theorems, like 'Goldbach's Theorem,' which have never been proved and which any fool could have guessed." Faber and..

An arithmetic progression of primes is a set of primes of the form for fixed and and consecutive , i.e., . For example, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a 10-term arithmetic progression of primes with difference 210.It had long been conjectured that there exist arbitrarily long sequences of primes in arithmetic progression (Guy 1994). As early as 1770, Lagrange and Waring investigated how large the common difference of an arithmetic progression of primes must be. In 1923, Hardy and Littlewood (1923) made a very general conjecture known as the k-tuple conjecture about the distribution of prime constellations, which includes the hypothesis that there exist infinitely long prime arithmetic progressions as a special case. Important additional theoretical progress was subsequently made by van der Corput (1939), who proved than there are infinitely many triples of primes in arithmetic progression, and Heath-Brown (1981),..

In 1913, Ramanujan asked if the Diophantineequation of second ordersometimes called the Ramanujan-Nagell equation, has any solutions other than , 4, 5, 7, and 15 (Schroeppel 1972, Item 31; Ramanujan 2000, p. 327; OEIS A060728). These correspond to , 3, 5, 11, and 181 (OEIS A038198). Nagell (1948) and Skolem et al. (1959) showed there are no solutions past , thus establishing Ramanujan's question in the negative.A generalization to two variables and was considered by Euler (Engel 1998, p. 126).

Fermat's last theorem is a theorem first proposed by Fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The scribbled note was discovered posthumously, and the original is now lost. However, a copy was preserved in a book published by Fermat's son. In the note, Fermat claimed to have discovered a proof that the Diophantine equation has no integer solutions for and .The full text of Fermat's statement, written in Latin, reads "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet" (Nagell 1951, p. 252). In translation, "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number..

The conjecture made by Belgian mathematician Eugène Charles Catalan in 1844 that 8 and 9 ( and ) are the only consecutive powers (excluding 0 and 1). In other words,(1)is the only nontrivial solution to Catalan'sDiophantine problem(2)The special case and is the case of a Mordell curve.Interestingly, more than 500 years before Catalan formulated his conjecture, Levi ben Gerson (1288-1344) had already noted that the only powers of 2 and 3 that apparently differed by 1 were and (Peterson 2000).This conjecture had defied all attempts to prove it for more than 150 years, although Hyyrő and Makowski proved that no three consecutive powers exist (Ribenboim 1996), and it was also known that 8 and 9 are the only consecutive cubic and square numbers (in either order). Finally, on April 18, 2002, Mihăilescu sent a manuscript proving the entire conjecture to several mathematicians (van der Poorten 2002). The proof has now appeared in..

The Mordell conjecture states that Diophantine equations that give rise to surfaces with two or more holes have only finite many solutions in Gaussian integers with no common factors (Mordell 1922). Fermat's equation has holes, so the Mordell conjecture implies that for each integer , the Fermat equation has at most a finite number of solutions.This conjecture was proved by Faltings (1984) and hence is now also known as Falting's theorem.

Archimedes' cattle problem, also called the bovinum problema, or Archimedes' reverse, is stated as follows: "The sun god had a herd of cattle consisting of bulls and cows, one part of which was white, a second black, a third spotted, and a fourth brown. Among the bulls, the number of white ones was one half plus one third the number of the black greater than the brown; the number of the black, one quarter plus one fifth the number of the spotted greater than the brown; the number of the spotted, one sixth and one seventh the number of the white greater than the brown. Among the cows, the number of white ones was one third plus one quarter of the total black cattle; the number of the black, one quarter plus one fifth the total of the spotted cattle; the number of spotted, one fifth plus one sixth the total of the brown cattle; the number of the brown, one sixth plus one seventh the total of the white cattle. What was the composition of the herd?"Solution consists..

The only Wiedersehen surfaces are the standard round spheres. The conjecture was proven by combining the Berger-Kazdan comparison theorem with A. Weinstein's results for even and C. T. Yang's for odd. Green (1963) obtained the first proof of the Blaschke's conjecture in the two-dimensional case.

Serre's problem, also called Serre's conjecture, asserts that the implication "free module projective module" can be reversed for every module over the polynomial ring , where is a field (Serre 1955).The hard part of the proof, the one concerning finitely generated modules, was given simultaneously, and independently, by D. Quillen in Cambridge, Massachusetts and A. A. Suslin in Leningrad (St. Petersburg) in 1976. As a result, the statement is often referred to as the "Quillen-Suslin theorem."The solution to this difficult problem is part of the work for which Quillen wasawarded the Fields Medal in 1978.Quillen and Suslin received, for other contributions in algebra, the ColePrize in 1975 and 2000 respectively.

In the directed graph above, pick any vertex and follow the arrows in sequence blue-red-red three times. You will finish at the green vertex. Similarly, follow the sequence blue-blue-red three times and you will always end on the yellow vertex, no matter where you started. This is called a synchronized coloring.The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with the same outdegree and where the greatest common divisor of all cycles lengths is 1. Trahtman (2007) provided a positive solution to this problem.

An acyclic digraph is a directed graph containing no directed cycles, also known as a directed acyclic graph or a "DAG." Every finite acyclic digraph has at least one node of outdegree 0. The numbers of acyclic digraphs on , 2, ... vertices are 1, 2, 6, 31, 302, 5984, ... (OEIS A003087).The numbers of labeled acyclic digraphs on , 2, ... nodes are 1, 3, 25, 543, 29281, ... (OEIS A003024). Weisstein's conjecture proposed that positive eigenvalued -matrices were in one-to-one correspondence with labeled acyclic digraphs on nodes, and this was subsequently proved by McKay et al. (2004). Counts for both are therefore given by the beautiful recurrence equationwith (Harary and Palmer 1973, p. 19; Robinson 1973, pp. 239-273).

Given a (0,1)-matrix, fill 11 spaces in each row in such a way that all columns also have 11 spaces filled. Furthermore, each pair of rows must have exactly one filled space in the same column. This problem is equivalent to finding a projective plane of order 10. Using a computer program, Lam et al. (1989) showed that no such arrangement exists.Lam's problem is equivalent to finding nine orthogonal Latinsquares of order 10.

Hadjicostas's formula is a generalization of the unitsquare double integral(1)(Sondow 2003, 2005; Borwein et al. 2004, p. 49), where is the Euler-Mascheroni constant. It states(2)for , where is the gamma function and is the Riemann zeta function (although care must be taken at because of the removable singularity present there). It was conjectured by Hadjicostas (2004) and almost immediately proved by Chapman (2004). The special case gives Beukers's integral for ,(3)(Beukers 1979). At , the formula is related to Beukers's integral for Apéry's constant , which is how interest in this class of integrals originally arose.There is an analogous formula(4)for , due to Sondow (2005), where is the Dirichlet eta function. This includes the special cases(5)(6)(7)(OEIS A094640; Sondow 2005) and(8)(9)(OEIS A103130), where is the Glaisher-Kinkelin constant (Sondow 2005)...

A power series in a variable is an infinite sum of the formwhere are integers, real numbers, complex numbers, or any other quantities of a given type.Pólya conjectured that if a function has a power series with integer coefficients and radius of convergence 1, then either the function is rational or the unit circle is a natural boundary (Pólya 1990, pp. 43 and 46). This conjecture was stated by G. Polya in 1916 and proved to be correct by Carlson (1921) in a result that is now regarded as a classic of early 20th century complex analysis.For any power series, one of the following is true: 1. The series converges only for . 2. The series converges absolutely for all . 3. The series converges absolutely for all in some finite open interval and diverges if or . At the points and , the series may converge absolutely, converge conditionally, or diverge. To determine the interval of convergence, apply the ratio test for absolute convergence..

A conjecture due to M. S. Robertson in 1936 which treats a univalent power series containing only odd powers within the unit disk. This conjecture implies the Bieberbach conjecture and follows in turn from the Milin conjecture. de Branges' proof of the Bieberbach conjecture proceeded by proving the Milin conjecture, thus establishing the Robertson conjecture and hence implying the truth of the Bieberbach conjecture.

If is a simple closed curve in , then the Jordan curve theorem, also called the Jordan-Brouwer theorem (Spanier 1966) states that has two components (an "inside" and "outside"), with the boundary of each.The Jordan curve theorem is a standard result in algebraic topology with a rich history. A complete proof can be found in Hatcher (2002, p. 169), or in classic texts such as Spanier (1966). Recently, a proof checker was used by a Japanese-Polish team to create a "computer-checked" proof of the theorem (Grabowski 2005).

In his monumental treatise Disquisitiones Arithmeticae, Gauss conjectured that the class number of an imaginary quadratic field with binary quadratic form discriminant tends to infinity with . A proof was finally given by Heilbronn (1934), and Siegel (1936) showed that for any , there exists a constant such thatas . However, these results were not effective in actually determining the values for a given of a complete list of fundamental discriminants such that , a problem known as Gauss's class number problem.Goldfeld (1976) showed that if there exists a "Weil curve" whose associated Dirichlet L-series has a zero of at least third order at , then for any , there exists an effectively computable constant such thatGross and Zaiger (1983) showed that certain curves must satisfy the condition of Goldfeld, and Goldfeld's proof was simplified by Oesterlé (1985)...

The number of digits in an integer is the number of numbers in some base (usually 10) required to represent it. The numbers 1 to 9 are therefore single digits, while the numbers 10 to 99 are double digits. Terms such as "double-digit inflation" are occasionally encountered, although this particular usage has thankfully not been needed in the U.S. for some time. The number of base- digits in a number can be calculated as(1)where is the floor function. For , the formula becomes(2)The number of digits in the number represented in base is given by the Wolfram Language function DigitCount[n, b, d], with DigitCount[n, b] giving a list of the numbers of each digit in . The total number of digits in a number is given by IntegerLength[n, b].The positive integers with distinct base-10 digits are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, ... (OEIS A010784). The number of -digit integers is given by(3)(4)(5)(6)where is..

The kissing number of a sphere is 12. This led Fejes Tóth (1943) to conjecture that in any unit sphere packing, the volume of any Voronoi cell around any sphere is at least as large as a regular dodecahedron of inradius 1. This statement is now known as the dodecahedral conjecture. It implies a bound of on the packing density for sphere packing, and thus provides a bound on the densest possible sphere packing. It is not, however, sufficient to establish the Kepler conjecture (which implies ).This long-outstanding conjecture was proved by Hales and McLaughlin (2002) using techniques of interval arithmetic and linear programming.

For a finite group , let be the subgroup generated by all the Sylow p-subgroups of . If is a projective curve in characteristic , and if , ..., are points of (for ), then a necessary and sufficient condition that occur as the Galois group of a finite covering of , branched only at the points , ..., , is that the quotient group has generators.Raynaud (1994) solved the Abhyankar problem in the crucial case of the affine line (i.e., the projective line with a point deleted), and Harbater (1994) proved the full Abhyankar conjecture by building upon this special solution.

The th coefficient in the power series of a univalent function should be no greater than . In other words, ifis a conformal mapping of a unit disk on any domain and and , then . In more technical terms, "geometric extremality implies metric extremality." An alternate formulation is that for any schlicht function (Krantz 1999, p. 150).The conjecture had been proven for the first six terms (the cases , 3, and 4 were done by Bieberbach, Lowner, and Garabedian and Schiffer, respectively), was known to be false for only a finite number of indices (Hayman 1954), and true for a convex or symmetric domain (Le Lionnais 1983). The general case was proved by Louis de Branges (1985). de Branges proved the Milin conjecture, which established the Robertson conjecture, which in turn established the Bieberbach conjecture (Stewart 1996).authorresultBieberbach (1916)Löwner (1923)Garabedian and Schiffer (1955)Pederson (1968), Ozawa..

Schmidt (1993) proposed the problem of determining if for any integer , the sequence of numbers defined by the binomial sums(1)are all integers.The following table gives the first few values of for small .OEISvalues1A0018501, 3, 13, 63, 321, 1683, 8989, 48639, ...2A0052591, 5, 73, 1445, 33001, 819005, ...3A0928131, 9, 433, 36729, 3824001, 450954009, ...4A0928141, 17, 2593, 990737, 473940001, ...5A0928151, 33, 15553, 27748833, 61371200001, ...This was proved by Strehl (1993, 1994) and Schmidt (1995) for the case , corresponding to the Franel numbers. Strehl (1994) also found an explicit expression for the case . The resulting identities for are therefore known as the Strehl identities. The problem was restated in Graham et al. (1994, pp. 256 and 549), who indicated that H. S. Wilf had shown to be an integer for any for (Zudilin 2004).The problem was answered in the affirmative by Zudilin (2004), who found explicit expressions..

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

Let a chess piece make a tour on an chessboard whose squares are numbered from 1 to along the path of the chess piece. Then the tour is called a magic tour if the resulting arrangement of numbers is a magic square, and a semimagic tour if the resulting arrangement of numbers is a semimagic square. If the first and last squares traversed are connected by a move, the tour is said to be closed (or "re-entrant"); otherwise it is open. (Note some care with terminology is necessary. For example, Jelliss terms a semimagic tour a "magic tour" and a magic tour a "diagonally magic tour.")Magic knight graph tours are not possible on boards for odd. However, as had long been known, they are possible for all boards of size for . However, the () remained open even since it was first investigated by authors such as Beverley (1848). It was not resolved until an exhaustive computer enumeration of all possibilities was completed on August 5,..

Macdonald's plane partition conjecture proposes a formula for the number of cyclically symmetric plane partitions (CSPPs) of a given integer whose Ferrers diagrams fit inside an box. Macdonald gave a product representation for the power series whose coefficients were the number of such partitions of .Let be the set of all integer points in the first octant such that a plane partition is defined and . Then is said to be cyclically symmetric if is invariant under the mapping . Let be the number of cyclically symmetric partitions of such that none of exceed . Let be the box containing all integer points such that , then is the number of cyclically symmetric plane partitions of such that . Now, let be the set of all the orbits in . Finally, for each point in , let its height(1)and for each in , let be the number of points in (either 1 or 3) and write(2)Then Macdonald conjectured that(3)(4)(5)(Mills et al. 1982, Macdonald 1995), where the latter form is due to Andrews(1979).The..

The "Foxtrot series" is a mathematical sum that appeared in the June 2, 1996 comic strip FoxTrot by Bill Amend (Amend 1998, p. 19; Mitchell 2006/2007). It arose from a convergence testing problem in a calculus book by Anton, but was inadvertently converted into a summation problem on an alleged final exam by the strip's author:(1)The sum can be done using partial fraction decomposition to obtain(2)(3)(4)(5)(OEIS A127198), where and the last sums have been done in terms of the digamma function and symbolically simplified.

A problem posed by the Slovak mathematician Stefan Znám in 1972 asking whether, for all integers , there exist integers all greater than 1 such that is a proper divisor of for each . The answer is negative for (Jának and Skula 1978) and affirmative for (Sun Qi 1983). Sun Qi also gave a lower bound for the number of solutions.All solutions for have now been computed, summarized in the table below. The numbers of solutions for , 3, ... terms are 0, 0, 0, 2, 5, 15, 93, ... (OEIS A075441), and the solutions themselves are given by OEIS A075461.known solutions references20--Jának and Skula (1978)30--Jának and Skula (1978)40--Jának and Skula (1978)522, 3, 7, 47, 3952, 3, 11, 23, 31652, 3, 7, 43, 1823, 1936672, 3, 7, 47, 403, 194032, 3, 7, 47, 415, 81112, 3, 7, 47, 583, 12232, 3, 7, 55, 179, 243237152, 3, 7, 43, 1807, 3263447, 2130014000915Jának and Skula (1978)2, 3, 7, 43, 1807, 3263591, 71480133827Cao, Liu, and Zhang..

In 1611, Kepler proposed that close packing (either cubic or hexagonal close packing, both of which have maximum densities of ) is the densest possible sphere packing, and this assertion is known as the Kepler conjecture. Finding the densest (not necessarily periodic) packing of spheres is known as the Kepler problem.Buckminster Fuller (1975) claimed to have a proof, but it was really a description of face-centered cubic packing, not a proof of its optimality (Sloane 1998). A second putative proof of the Kepler conjecture was put forward by W.-Y. Hsiang (Cipra 1991, Hsiang 1992, 1993, Cipra 1993), but was subsequently determined to be flawed (Conway et al. 1994, Hales 1994, Sloane 1998). According to J. H. Conway, nobody who has read Hsiang's proof has any doubts about its validity: it is nonsense.Soon thereafter, Hales (1997a) published a detailed plan describing how the Kepler conjecture might be proved using a significantly..

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).

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