Portal:Mathematics
Mathematics is the study of numbers, quantity, space, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.
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There are approximately 31,444 mathematics articles in Wikipedia.
The real part (red) and imaginary part (blue) of the critical line Re(s) = 1/2 of the Riemann zetafunction. Image credit: User:Army1987 
The Riemann hypothesis, first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. It has been an open question for well over a century, despite attracting concentrated efforts from many outstanding mathematicians.
The Riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zetafunction ζ(s). The Riemann zetafunction is defined for all complex numbers s ≠ 1. It has zeros at the negative even integers (i.e. at s=2, s=4, s=6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the nontrivial zeros, and states that:
 The real part of any nontrivial zero of the Riemann zeta function is ½
Thus the nontrivial zeros should lie on the socalled critical line ½ + it with t a real number and i the imaginary unit. The Riemann zetafunction along the critical line is sometimes studied in terms of the Zfunction, whose real zeros correspond to the zeros of the zetafunction on the critical line.
The Riemann hypothesis is one of the most important open problems in contemporary mathematics; a $1,000,000 prize has been offered by the Clay Mathematics Institute for a proof. Most mathematicians believe the Riemann hypothesis to be true. (J. E. Littlewood and Atle Selberg have been reported as skeptical. Selberg's skepticism, if any, waned, from his young days. In a 1989 paper, he suggested that an analogue should hold for a much wider class of functions, the Selberg class.)
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Here a polyhedron called a truncated icosahedron (left) is compared to the classic Adidas Telstar–style football (or soccer ball). The familiar 32panel ball design, consisting of 12 black pentagonal and 20 white hexagonal panels, was first introduced by the Danish manufacturer Select Sport, based loosely on the geodesic dome designs of Buckminster Fuller; it was popularized by the selection of the Adidas Telstar as the official match ball of the 1970 FIFA World Cup. The polyhedron is also the shape of the Buckminsterfullerene (or "Buckyball") carbon molecule initially predicted theoretically in the late 1960s and first generated in the laboratory in 1985. Like all polyhedra, the vertices (corner points), edges (lines between these points), and faces (flat surfaces bounded by the lines) of this solid obey the Euler characteristic, V − E + F = 2 (here, 60 − 90 + 32 = 2). The icosahedron from which this solid is obtained by truncating (or "cutting off") each vertex (replacing each by a pentagonal face), has 12 vertices, 30 edges, and 20 faces; it is one of the five regular solids, or Platonic solids—named after Plato, whose school of philosophy in ancient Greece held that the classical elements (earth, water, air, fire, and a fifth element called aether) were associated with these regular solids. The fifth element was known in Latin as the "quintessence", a hypothesized uncorruptible material (in contrast to the other four terrestrial elements) filling the heavens and responsible for celestial phenomena. That such idealized mathematical shapes as polyhedra actually occur in nature (e.g., in crystals and other molecular structures) was discovered by naturalists and physicists in the 19th and 20th centuries, largely independently of the ancient philosophies.
 ... that, while the crisscross algorithm visits all eight corners of the Klee–Minty cube when started at a worst corner, it visits only three more corners on average when started at a random corner?
 ...that in senary, all prime numbers other than 2 and 3 end in 1 or a 5?
 ... if the integer n is prime, then the nth Perrin number is divisible by n?
 ...that it is impossible to trisect a general angle using only a ruler and a compass?
 ...that in a group of 23 people, there is a more than 50% chance that two people share a birthday?
 ...that statistical properties dictated by Benford's Law are used in auditing of financial accounts as one means of detecting fraud?
 ...the hyperbolic trigonometric functions of the natural logarithm can be represented by rational algebraic fractions?
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