OEIS

Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime. Indeed, the first five Fermat numbers F0,...,F4 are easily shown to be prime. However, this conjecture was refuted by Leonhard Euler in 1732 when he showed that It is interesting to note how Euler found this factorization. Euler had proved that every factor of Fn must have the form k2n+1 + 1. For n = ...

See also:

Fermat number, Fermat number - Basic properties, Fermat number - Primality of Fermat numbers, Fermat number - Factorisation of Fermat numbers, Fermat number - Fermat's little theorem and pseudoprimes, Fermat number - Other theorems about Fermat's primes, Fermat number - Relationship to constructible polygons, Fermat number - Applications of Fermat numbers, Fermat number - Other interesting facts, Fermat number - Generalised Fermat numbers

Read more here: » Fermat number: Encyclopedia II - Fermat number - Primality of Fermat numbers

Factorial - Primorial. The primorial is similar to the factorial, but with the product taken only over the prime numbers. Factorial - Multifactorials. A common related notation is to use multiple exclamation points to denote a multifactorial, the product of integers in steps of two (n!!), three (n!!!), or more. n!! denotes the double factorial of nSee also:

Factorial, Factorial - Definition, Factorial - Non-integer factorials, Factorial - Applications, Factorial - Rate of growth, Factorial - Computation, Factorial - Factorial-like products, Factorial - Primorial, Factorial - Multifactorials, Factorial - Hyperfactorials, Factorial - Superfactorials, Factorial - Superfactorials alternative definition

Read more here: » Factorial: Encyclopedia II - Factorial - Factorial-like products

Some regular polygons are easy to construct with compass and straightedge; others are not. This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not, which n-gons are constructible and which are not? Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae. This theory allowed him to formulate a sufficient condition for the c ...

See also:

Constructible polygon, Constructible polygon - Conditions for constructibility, Constructible polygon - General theory, Constructible polygon - Detailed results in terms of Fermat primes, Constructible polygon - Compass-and-straightedge constructions, Constructible polygon - Other constructions

Read more here: » Constructible polygon: Encyclopedia II - Constructible polygon - Conditions for constructibility

If Q(x) denotes the number of square-free integers between 1 and x, then (see pi and big O notation). The asymptotic/natural density of square-free numbers is therefore where ζ is the Riemann zeta function. Likewise, if Q(x,n) denotes the number of nth power-free integers between 1 and x, one can show ...

See also:

Square-free integer, Square-free integer - Equivalent characterizations of square-free numbers, Square-free integer - Distribution of square-free numbers, Square-free integer - Erdös Squarefree Conjecture

Read more here: » Square-free integer: Encyclopedia II - Square-free integer - Distribution of square-free numbers

Although mathematicians regard the Riemann zeta function as being primarily relevant to the "purest" of mathematical disciplines, number theory, it also occurs in applied statistics (see Zipf's law and Zipf-Mandelbrot law), physics, and the mathematical theory of musical tuning. During several physics-related calculations, one must evaluate the sum of the positive integers; paradoxically, on physical grounds one expects a finite answer. When this situation arises, there is typically a rigorous approach involving much in-depth analysis ...

See also:

Riemann zeta function, Riemann zeta function - Definition, Riemann zeta function - Values at the integers, Riemann zeta function - Relationship to prime numbers, Riemann zeta function - Proving the Euler product formula, Riemann zeta function - An easier proof for the layperson, Riemann zeta function - The importance of the zeros of ζs, Riemann zeta function - Basic properties, Riemann zeta function - The Riemann zeta function as a Mellin transform, Riemann zeta function - Series expansions, Riemann zeta function - Globally convergent series, Riemann zeta function - Universality, Riemann zeta function - Applications, Riemann zeta function - Generalizations, Riemann zeta function - Zeta functions in fiction

Read more here: » Riemann zeta function: Encyclopedia II - Riemann zeta function - Applications

The first approximation of this number was given in ancient Indian mathematical texts, the Sulbasutras (800 B.C. to 200 B.C.) as follows: Increase a unit length by its third and this third by its own fourth less the thirty-fourth part of that fourth. That is, The discovery of the irrational numbers is usually attributed to Pythagoras or one of his followers, who produced a (most likely ...

See also:

Square root of 2, Square root of 2 - History, Square root of 2 - Proof of irrationality, Square root of 2 - A different proof

Read more here: » Square root of 2: Encyclopedia II - Square root of 2 - History

The popular Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square. Sudoku imposes the additional restriction that 3×3 subgroups must also contain the digits 1–9 (in the standard version). The Diamond 16 Puzzle illustrates a generalized concept of Latin-square orthogonality: that of "orthogonal squares" (Diamond Theory, 1976) or "orthogonal matrices"-- orthogonal, that is, in a combinatorial, not a linear-algebra sense (A. E. Brouwer, ...

See also:

Latin square, Latin square - Orthogonal array representation, Latin square - Equivalence classes of Latin squares, Latin square - The number of Latin squares, Latin square - Examples, Latin square - Latin squares and mathematical puzzles

Read more here: » Latin square: Encyclopedia II - Latin square - Latin squares and mathematical puzzles

The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices which belong to one of 7 crystal systems. This results in a space group being some combination the translational symmetry of a unit cell including lattice centering, and the point group symmetry operations of reflection, rotation and rotoinversion (also called improper rotation). Furthermore one must consider the screw axis and glide plane symmetry operations. These are called compound symmetry operatioans and ...

See also:

Space group, Space group - Space groups in crystallography, Space group - Glide planes and screw axes, Space group - Notation, Space group - Group theory, Space group - Space groups in various dimensions, Space group - Grouping space groups by point group, Space group - Further categorizing of space groups

Read more here: » Space group: Encyclopedia II - Space group - Space groups in crystallography

The Riemann zeta function is meromorphic with a single pole of order one at s = 1. It can therefore be expanded as a Laurent series about s = 1; the series development then is The constants here are called the Stieltjes constants and can be defined as The constant term γ0 is the Euler-Mascheroni constant. Another series development valid for the entire complex plane is where is the rising factorial . This can be used recursively to ext ...

See also:

Riemann zeta function, Riemann zeta function - Definition, Riemann zeta function - Values at the integers, Riemann zeta function - Relationship to prime numbers, Riemann zeta function - Proving the Euler product formula, Riemann zeta function - An easier proof for the layperson, Riemann zeta function - The importance of the zeros of ζs, Riemann zeta function - Basic properties, Riemann zeta function - The Riemann zeta function as a Mellin transform, Riemann zeta function - Series expansions, Riemann zeta function - Globally convergent series, Riemann zeta function - Universality, Riemann zeta function - Applications, Riemann zeta function - Generalizations, Riemann zeta function - Zeta functions in fiction

Read more here: » Riemann zeta function: Encyclopedia II - Riemann zeta function - Series expansions

The zeta function satisfies the following functional equation: valid for all s in C\{0,1}. Here, Γ denotes the gamma function. This formula is used to construct the analytic continuation in the first place. At s = 1, the zeta function has a simple pole with residue 1. There is also a symmetric version of the functional equation, given by first defining The functional equation is then given by ξ(s) = ξ(1 − s). Euler was also able to calculate ζ(2k) for even ...

See also:

Riemann zeta function, Riemann zeta function - Definition, Riemann zeta function - Values at the integers, Riemann zeta function - Relationship to prime numbers, Riemann zeta function - Proving the Euler product formula, Riemann zeta function - An easier proof for the layperson, Riemann zeta function - The importance of the zeros of ζs, Riemann zeta function - Basic properties, Riemann zeta function - The Riemann zeta function as a Mellin transform, Riemann zeta function - Series expansions, Riemann zeta function - Globally convergent series, Riemann zeta function - Universality, Riemann zeta function - Applications, Riemann zeta function - Generalizations, Riemann zeta function - Zeta functions in fiction

Read more here: » Riemann zeta function: Encyclopedia II - Riemann zeta function - Basic properties

Main article: History of Pi. The value of π has been known in some form since antiquity. As early as the 20th century BC, Babylonian mathematicians were using π=25/8, which is within 0.5% of the exact value. It is sometimes claimed that the Bible states that π=3, based on a passage in 1 Kings 7:23 giving measurements for a round basin. Rabbi Nehemiah explained this by the diameter being from outside to outside while the circumference was the inner brim; but it may suffice that the measurements are given in round numbers. (Also, the basin ...

See also:

Pi, Pi - Properties, Pi - Formulae involving π, Pi - Geometry, Pi - Analysis, Pi - Continued fractions, Pi - Number theory, Pi - Dynamical systems and ergodic theory, Pi - Physics, Pi - Probability and statistics, Pi - History of π, Pi - Numerical approximations of π, Pi - Miscellaneous formulae, Pi - Less accurate approximations, Pi - Open questions, Pi - The nature of π, Pi - Fictional references, Pi - π culture, Pi - Memorizing Pi

Read more here: » Pi: Encyclopedia II - Pi - History of π

Pi - Geometry. π appears in many formulae in geometry involving circles and spheres. (All of these are a consequence of the first one, as the area of a circle can be written as A = ∫(2πr)dr ("sum of annuli of infinitesimal width"), and others concern a surface or solid of revolution.) Also, the angle measure of 180° (degrees) is equal to π radians. ...

See also:

Pi, Pi - Properties, Pi - Formulae involving π, Pi - Geometry, Pi - Analysis, Pi - Continued fractions, Pi - Number theory, Pi - Dynamical systems and ergodic theory, Pi - Physics, Pi - Probability and statistics, Pi - History of π, Pi - Numerical approximations of π, Pi - Miscellaneous formulae, Pi - Less accurate approximations, Pi - Open questions, Pi - The nature of π, Pi - Fictional references, Pi - π culture, Pi - Memorizing Pi

Read more here: » Pi: Encyclopedia II - Pi - Formulae involving π

The Mellin transform of a function f(x) is defined as in the region where the integral is defined. There are various expressions for the zeta function as a Mellin transform. If the real part of s is greater than one, we have By subtracting off the first terms of the power series expansion of 1/(exp(x) − 1) around zero, we can get the zeta function in other regions. In particular, in the critical strip we have < ...

See also:

Riemann zeta function, Riemann zeta function - Definition, Riemann zeta function - Values at the integers, Riemann zeta function - Relationship to prime numbers, Riemann zeta function - Proving the Euler product formula, Riemann zeta function - An easier proof for the layperson, Riemann zeta function - The importance of the zeros of ζs, Riemann zeta function - Basic properties, Riemann zeta function - The Riemann zeta function as a Mellin transform, Riemann zeta function - Series expansions, Riemann zeta function - Globally convergent series, Riemann zeta function - Universality, Riemann zeta function - Applications, Riemann zeta function - Generalizations, Riemann zeta function - Zeta functions in fiction

Read more here: » Riemann zeta function: Encyclopedia II - Riemann zeta function - The Riemann zeta function as a Mellin transform

π is an irrational number; that is, it cannot be written as the ratio of two integers, as was proven in 1761 by Johann Heinrich Lambert. π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with ruler and compass are constructible numbers, it is impossible to square the circle, that is, it is impossible to construct, using ruler and compass alone, a square whose area is equ ...

See also:

Pi, Pi - Properties, Pi - Formulae involving π, Pi - Geometry, Pi - Analysis, Pi - Continued fractions, Pi - Number theory, Pi - Dynamical systems and ergodic theory, Pi - Physics, Pi - Probability and statistics, Pi - History of π, Pi - Numerical approximations of π, Pi - Miscellaneous formulae, Pi - Less accurate approximations, Pi - Open questions, Pi - The nature of π, Pi - Fictional references, Pi - π culture, Pi - Memorizing Pi

Read more here: » Pi: Encyclopedia II - Pi - Properties

There are a number of related zeta functions that can be considered to be generalizations of Riemann's zeta. The simplest of these are the Hurwitz zeta function , which coincides with Riemann's zeta when q = 1. The polylogarithm is given by which coincides with Riemann's zeta when z = 1. The Lerch transcendent is given by which coincide ...

See also:

Riemann zeta function, Riemann zeta function - Definition, Riemann zeta function - Values at the integers, Riemann zeta function - Relationship to prime numbers, Riemann zeta function - Proving the Euler product formula, Riemann zeta function - An easier proof for the layperson, Riemann zeta function - The importance of the zeros of ζs, Riemann zeta function - Basic properties, Riemann zeta function - The Riemann zeta function as a Mellin transform, Riemann zeta function - Series expansions, Riemann zeta function - Globally convergent series, Riemann zeta function - Universality, Riemann zeta function - Applications, Riemann zeta function - Generalizations, Riemann zeta function - Zeta functions in fiction

Read more here: » Riemann zeta function: Encyclopedia II - Riemann zeta function - Generalizations

Gauss conjectured that an even better approximation to π(x) is given by the offset logarithmic integral function Li(x), defined by Indeed, this integral is strongly suggestive of the notion that the 'density' of primes around t should be 1/lnt. This function is related to the logarithm by the asymptotic expansion So, the prime number theorem can also be written as π(x) ~ Li(x). The advantage of this formulation is that the error term is ...

See also:

Prime number theorem, Prime number theorem - Statement of the theorem, Prime number theorem - The prime counting function in terms of the logarithmic integral, Prime number theorem - The issue of depth, Prime number theorem - The prime number theorem for arithmetic progressions, Prime number theorem - Bounds on the prime counting function, Prime number theorem - Approximations for the nth prime number, Prime number theorem - Gaps between primes, Prime number theorem - Table of πx x / ln x and Lix, Prime number theorem - Analogue for irreducible polynomials over a finite field

Read more here: » Prime number theorem: Encyclopedia II - Prime number theorem - The prime counting function in terms of the logarithmic integral

Richard Ewen Borcherds' proof of the conjecture of Conway and Norton can be broken into five major steps as follows: A vertex algebra V is constructed that is a graded algebra affording the moonshine representations on M, and it is verified that the monster module has a vertex algebra structure invariant under the action of M. V is thus called the Monster vertex algebra. A Lie algebra is constructed from V using the Goddard-Thorn "no-ghost" theorem from string theory; this is a genera ...

See also:

Monstrous moonshine, Monstrous moonshine - Formal versions of Conway's and Norton's conjectures, Monstrous moonshine - The Monster module, Monstrous moonshine - Borcherds' proof, Monstrous moonshine - Why monstrous moonshine?

Read more here: » Monstrous moonshine: Encyclopedia II - Monstrous moonshine - Borcherds' proof

The puzzle was designed by Howard Garns, a retired architect and freelance puzzle constructor, and first published in 1979. Although likely inspired by the Latin square invention of Leonhard Euler, Garns added a third dimension (the regional restriction) to the mathematical construct and (unlike Euler) presented the creation as a puzzle, providing a partially-completed grid and requiring the solver to fill in the rest. The puzzle was first published in New York by the specialist puzzle publisher Dell Magazines in its magazine Dell Pencil Puzzles and Word Games, under the title Number Place (which w ...

See also:

Sudoku, Sudoku - Introduction, Sudoku - Rules and terminology, Sudoku - Solution methods, Sudoku - Scanning, Sudoku - Marking up, Sudoku - Analysis, Sudoku - Computer solutions, Sudoku - Difficulty ratings, Sudoku - Construction, Sudoku - Variants, Sudoku - Mathematics of Sudoku, Sudoku - History, Sudoku - Popularity in the media

Read more here: » Sudoku: Encyclopedia II - Sudoku - History

Simon Plouffe gives the identities and ...

See also:

Zeta constants, Zeta constants - ζ3, Zeta constants - ζ5, Zeta constants - ζ7, Zeta constants - ζ2n+1, Zeta constants - ζ2n

Read more here: » Zeta constants: Encyclopedia II - Zeta constants - ζ5

All numbers in base 10 with one digit {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} are palindromic ones. The number of palindromic numbers with two digits is 9: {11, 22, 33, 44, 55, 66, 77, 88, 99}. There are 90 palindromic numbers with three digits: {101, 111, 121, 131, 141, 151, 161, 171, 181, 191, ..., 909, 919, 929, 939, 949, 959, 969, 979, 989, 999} and also 90 palindromic numbers with four digits: {1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, ..., 9 ...

See also:

Palindromic number, Palindromic number - Formal definition, Palindromic number - Decimal palindromic numbers, Palindromic number - Other bases

Read more here: » Palindromic number: Encyclopedia II - Palindromic number - Decimal palindromic numbers

Palindromic numbers can be considered in other numeral systems than decimal. For example, the binary palindromic numbers are: 0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, … or in decimal: 0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, … (sequence A006995 in OEIS). The Mersenne primes form a subset of the binary palindromic primes. Generally, a number that is palindromic in one base is not palindromic in another base; for instance, 1646110 = 404D< ...

See also:

Palindromic number, Palindromic number - Formal definition, Palindromic number - Decimal palindromic numbers, Palindromic number - Other bases

Read more here: » Palindromic number: Encyclopedia II - Palindromic number - Other bases

The general problem of solving Sudoku puzzles on n2 x n2 boards of n x n blocks is known to be NP-complete [6]. This gives some indication of why Sudoku is difficult to solve, although on boards of finite size the problem is finite and can be solved by a deterministic finite automaton that knows the entire game tree. Solving Sudoku puzzles can be expressed as a graph colouring problem. The aim of the puzzle in its standard form is to construct a proper 9-colouring ...

See also:

Sudoku, Sudoku - Introduction, Sudoku - Rules and terminology, Sudoku - Solution methods, Sudoku - Scanning, Sudoku - Marking up, Sudoku - Analysis, Sudoku - Computer solutions, Sudoku - Difficulty ratings, Sudoku - Construction, Sudoku - Variants, Sudoku - Mathematics of Sudoku, Sudoku - History, Sudoku - Popularity in the media

Read more here: » Sudoku: Encyclopedia II - Sudoku - Mathematics of Sudoku