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...' be any [[natural number]]. Wilson's theorem says that ''n'' is a [[prime number]] if and only if: ...uation is correct. Also, if the equation is correct, then ''n'' is a prime number. The equation says that the [[factorial]] of ''(n - 1)'' is one less than a ...

...bout the properties of [[Prime number|prime]]s. It says that if ''a'' is a number, and ''p'' is a prime, then [[Category:Number theory]] ...

In [[number theory]] a '''Carmichael number''' is a [[composite number|composite]] positive [[integer]] <math>n</math>, which satisfies the [[cong ...are composite numbers that behave a little bit like they would be a prime number. ...

...]], the '''totient''' of a [[positive number|positive]] [[integer]] is the number of integers smaller than ''n'' which are [[coprime]] to ''n'' (they share n ...}/n\mathbb{Z}</math>. This fact, together with [[Lagrange's theorem (group theory)|Lagrange's theorem]], provides a proof for [[Euler's theorem]]. ...

...''x''. It is written as <math>\pi(x)</math>, but it is not related to the number [[Pi (mathematical constant)|π]]. [[Category:Number theory]] ...

A [[real number|real]] or [[complex number]] is called a '''transcendental number''' if it cannot be found as a result of an algebraic equation with [[intege ...ental can be very hard. Each transcendental number is also an [[irrational number]]. The first people to see that there were transcendental numbers were [[Go ...

...ber]]. [[Cryptography]] uses prime numbers, and needs to test if a certain number is prime. The official proof of a prime is through its primality certifica ...ve of Eratosthenes]]: Determine if there is a number (between 2 and n, the number to test) that divides n, without a rest ...

...) cardinality if they have the same number of elements. They have the same number of elements [[if and only if]] there is a [[1-to-1 correspondence]] between ...set is only one way of giving a number to the ''size'' of a set. [[Measure theory|Measure]] is different. ...

'''Cardinal numbers''' (or '''cardinals''') are [[number]]s that say ''how many'' of something there are, for example: one, two, thr ...''[[Set#Cardinality_of_a_set|cardinality]]''' of a [[set]] is the cardinal number that tells how many things are in the set. ...

...arithmetic]], a [[number]] g is a '''primitive root modulo n''', if every number m from 1..(n-1) can be expressed in the form of <math>g^x\equiv m \pmod n</ ..., 2, \ldots, 6</math> of the group modulo 7 can be expressed that way. The number 2 is no primitive root modulo 7, because ...

* 概率论（Probability Theory） # [[组合数学 (Spring 2014)/Pólya's theory of counting|Pólya's theory of counting]] ...

...ies. Later, in the early 90s, [[Ralph Greenberg]] has suggested an Iwasawa theory for [[motive (algebraic geometry)|motives]]. ...ete group of all <math> p</math>-power [[roots of unity]] in the [[complex number]]s. ...

* 概率论（Probability Theory） # [[组合数学 (Spring 2013)/Pólya's theory of counting|Pólya's theory of counting]] | [http://tcs.nju.edu.cn/slides/comb2013/comb6.pdf slides1] | ...

* 概率论（Probability Theory） # [[组合数学 (Spring 2015)/Pólya's theory of counting|Pólya's theory of counting | Pólya计数法]]（ [http://tcs.nju.edu.cn/slides/comb2015/PolyaTheor ...

In [[graph theory]] a '''graph''' is used to represent the connection between two [[set]]s. O ...t, where an edge that connects to the vertex at both ends (a [[loop (graph theory)|loop]]) is counted twice. ...

...ves.<ref>Negative numbers have a [[:wikt:minus|minus]] (−) in front of the number. Positive numbers have no sign or a [[:wikt:plus|plus]] (+) sign in front. ...rational number]] with no "fraction", or part. An integer is a [[decimal]] number with all zeros after the [[decimal separator]]. (For example, the integer 1 ...

...bottom of the time signature can be any exponent of 2. So, 64 could be a number that is put in the bottom of the time signature, but 65 could not be one.<r !Number on the bottom of the time signature ...

* 概率论（Probability Theory） # [[组合数学 (Fall 2017)/Pólya's theory of counting|Pólya's theory of counting | Pólya计数法]] （ [http://tcs.nju.edu.cn/slides/comb2017/Polya.pdf ...

* 概率论（Probability Theory） # [[组合数学 (Fall 2011)/Pólya's theory of counting|Pólya's theory of counting]] ...

...is a proper subset of the set of [[real number]]s or the set of [[positive number]]s. ...,1264} is its own subset, and it's a proper subset of the set of [[natural number]]s. ...

* 概率论（Probability Theory） # [[组合数学 (Fall 2019)/Pólya's theory of counting|Pólya's theory of counting | Pólya计数法]] （ [http://tcs.nju.edu.cn/slides/comb2019/Polya.pdf ...

The set of [[natural number]]s is '''not''' an alphabet because as it is not finite. * Michael A. Harrison, ''Introduction to Formal Language Theory'', (1978): Chapter 1.2 ...

In [[mathematics]], a '''Mersenne number''' is a number that is one less than a ''power of two''. ...the definition of a Mersenne number where exponent ''n'' has to be a prime number. ...

In [[mathematics]], a '''Ramanujan prime''' is a [[prime number]] that satisfies a result proven by [[Srinivasa Ramanujan]]. It relates to ...)'' is the [[prime counting function]]. The prime counting function is the number of primes less than or equal to ''x''. ...

...any elements are in H, called the order of H) divides |G|. Moreover, the number of distinct left (right) [[coset]]s of H in G is |G|/|H|. ...

* 概率论（Probability Theory） # [[组合数学 (Fall 2023)/Pólya's theory of counting|Pólya's theory of counting | Pólya计数法]] ([http://tcs.nju.edu.cn/slides/comb2023/Polya.pdf ...

...ss the test, and that are not prime. These numbers are called [[Carmichael number]]s. ...the equality does not hold for a value of ''a'', then ''n'' is [[composite number|composite]] (not prime). If the equality does hold for many values of ''a' ...

...ors are for values close to zero and [[hue]] encodes the value's [[complex number|argument]]. ...med after [[Bernhard Riemann]], who wrote about it in the memoir "[[On the Number of Primes Less Than a Given Quantity]]", published in 1859. ...

A '''Fermat number''' is a special [[positive number|positive]] [[number]]. Fermat numbers are named after [[Pierre de Fermat]]. The formula that ge ...a power of two. Every prime of the form 2^''n'' + 1 is a Fermat number, and such primes are called '''Fermat primes'''. The only known Fermat prim ...

* 概率论（Probability Theory） # [[Combinatorics (Fall 2010)/Extremal set theory|Extremal set theory]] | [http://lamda.nju.edu.cn/yinyt/notes/comb2010/comb8.pdf slides] ...

...eory]]. He proved that the answer to his problem was smaller than Graham's number. ...ers ever used in a [[mathematical proof]]. Even if every digit in Graham's number were written in the tiniest writing possible, it would still be too big to ...

* 概率论（Probability Theory） # [[组合数学 (Fall 2024)/Pólya's theory of counting|Pólya's theory of counting | Pólya计数法]] ([http://tcs.nju.edu.cn/slides/comb2024/Polya.pdf ...

...contributed to many areas of learning. Most of his work was about [[number theory]] and [[astronomy]]. ...her once tried to keep the children busy, telling them to add up all the [[number]]s from 1 to 100. Gauss did it quickly, like this: 1 + 100 = 101, 2 + 99 = ...

...ame way that all real numbers can be thought of as 1 multiplied by another number. Arithmetic functions such as, [[addition]], subtraction, [[multiplication] ...o pose real problems for mathematicians. As a comparison, using [[negative number]]s, it is possible to find the ''x'' in the [[equation]] <math>a + x = b</m ...

...82845904523536, come close to the true value. The true value of ''e'' is a number that never ends. Euler himself gave the first 23 digits of e.<ref>[[Leonhar ...onential function]]s. For example, the exponential function applied to the number one, has a value of ''e''. ...

...the complex plane. The real part <math>\operatorname{Re}(s)</math> of the number is drawn horizontally, the imaginary part <math>\operatorname{Im}(s)</math> ...rom Riemann's functional equation. More have been computed and have [[real number|real]] part 1/2. The hypothesis states all the undiscovered zeros must hav ...

...argument ''x'' fixed at 10. In [[number theory]], the concept of [[p-adic number]]s is also closely related to that of a power series. ...

...as there is no real number that will multiply by itself to get a negative number (e.g. 3*3 = 9 and -3*-3 = 9). ...mbers is a lot like multiplying a '''positive number''' with a '''negative number'''. If I say "go east by 2*-3 miles" it means "rotate all of the way around ...

...ction of two numbers can produce a negative number, which is not a natural number, but it is an integer. ...osure of [[intersection]]. Open sets have infinite closure of [[union (set theory)|union]]. That is, in mathematical notation, if A₀, and A<sub>1< ...

Mathematics is the study of [[number]]s, [[shape]]s and [[pattern]]s. The word is sometimes shortened to '''mat === Number === ...

...set of elements is the set of prime [[divisor]]s of ''n''. For example the number ''120'' has the [[factorisation|prime factorisation]] ...example, has two solutions. However, in some cases they are both the same number. Thus the multiset of solutions of the equation could be { 3, 5 }, or it c ...

...nal]] [[number]]. Usually when people say "number" they usually mean "real number". The official symbol for real numbers is a bold '''R''' or a blackboard bo ...an infinitely long [[ruler]]. There is a mark for [[zero]] and every other number, in order of size. Unlike a ruler, there are numbers below zero. These are ...

...also worked with the number, and tried, unsuccessfully, to approximate the number to 32 decimal places, making mistakes on five digits. <ref>{{cite web|last= ...function]], which gives the greatest integer less than or equal to a given number. ...

...nd so is a set with one hundred things in it. A set with all the [[natural number]]s (counting numbers) in it is countable too. It's [[infinite]] but if som Countable sets include all sets with a [[finite set|finite]] number of members, no matter how many. ...

...series of special numbers: 2, 1729 etc. A taxicab number is the smallest number that can be expressed as the sum of two positive cubes in ''n'' distinct wa ...n]], who was ill. Both men were mathematicians and liked to think about [[number]]s. ...

...search problems in graphs, and has theoretical significance in complexity theory. ...the current position, and another <math>O(\log n)</math> bits to count the number of steps. So the total space used by the algorithm inaddition to the input ...

...have opposite [[charge]] and properties, such as [[lepton]] and [[baryon]] number. ...rgy a given piece of something has. Since the speed of light is such a big number, this means that even a small amount of matter can have a lot of energy (it ...

...nction is said to be '''discontinuous'''. Functions defined on the [[real number]]s, with one input and one output variable, will show as an ''uninterrupted ...continuity: Suppose that there is a function ''f'', defined on the [[real number]]s. At the point <math>x_0</math> the function will have the value <math>f( ...

...<math>\left \{ a^{n}\right\}</math>, where <math>n\,</math> is a [[natural number]] and <math>a^n\,</math> means <math>a\,</math> repeated <math>n\,</math> t | title = Introduction to Automata Theory, Languages, and Computation ...

...of some [[formal language]] a unique [[natural number]] called a '''Gödel number (GN)'''. The concept was first used by [[Kurt Gödel]] for the proof of his ...]]s can then represent some form or function. A [[numbering (computability theory)|numbering]] of the set of [[computable function]]s can then be represented ...

A '''Wilson prime''' is a special kind of [[prime number]]. A prime number ''p'' is a Wilson prime if (and only if [ [[iff]] ]) ...is a [[positive number|positive]] [[integer]] (sometimes called [[natural number]]). Wilson primes were first described by [[Emma Lehmer]].<ref>[http://www. ...

...stand how anything can be both a wave and a particle, scientists do have a number of [[equation]]s for describing these things that have variables for both [ == Basic theory == ...

The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one ...\mathbb{R}</math> is a function that maps each event to a nonnegative real number, satisfying ...

* {''x'' | ''x'' is a natural number & ''x'' < 4}. ...poken English, that is: "the set of all ''x'' such that ''x'' is a natural number and ''x'' is less than four". ...

...bility theory]] and [[statistics]], the '''median''' is a [[number]]. This number has the property that it divides a set of observed values in two equal halv ...n, at position 7. With an [[even number]] of values, as there is no single number which divides all of the numbers to two halves, the median is defined as th ...

...umber)|15]] [[16 (number)|16]] [[17 (number)|17]] [[18 (number)|18]] [[19 (number)|19]] &nbsp; [[twenty|20]] &nbsp; ...umber)|25]] [[26 (number)|26]] [[27 (number)|27]] [[28 (number)|28]] [[29 (number)|29]] &nbsp; [[thirty|30]] &nbsp; ...

...s small space as possible. We hope for a small space storing only constant number of items. (Interviewing problem of Theory Group in Microsoft Research Asia for student interns.) ...

...ath> satisfying the condition in the Ramsey theory is called the '''Ramsey number'''. Prove that: ...no red triangles and no blue <math>4</math>-cliques. Try to determine the number of red and blue edges adjacent to each vertex.) ...

[[File:Minkowski diagram - asymmetric.svg|thumb|In the theory of relativity both observers assign the event at A to different times.]] ...a spatial dimension, (the y-axis); unfortunately, this is the limit to the number of dimensions: graphing in four dimensions is impossible. The rule for grap ...

...nction]] whose value is [[0 (number)|zero]] for negative argument and [[1 (number)|one]] for positive argument. The function is used in the mathematics of [[control theory]] to represent a signal that switches on at a specified time and stays swit ...

Normal [[hotel]]s have a set number of rooms. This number is [[finite]]. Once every room has been assigned to a guest, any new guest ...s already full, something that could not happen in any hotel with a finite number of rooms. ...

...ne by the Babylonians; Ancient Greeks, like [[Ptolemy]] later extended the theory. The third number is the height, altitude, or depth. This is given with respect to some fixed ...

...ers rolls a (fair) 6-sided die. For any pair of players who throw the same number, the group scores <math>1</math> point. Find the mean and variance of the t == Problem 3 (Probability meets graph theory) == ...

...ord.</ref>^p160 It is a central idea in [[evolution|evolutionary theory]]. Fitness is usually equal to the proportion of the individual's [[gene]]s ...or to help relatives with similar genes to reproduce, ''as long as similar number of copies of individual's genes get passed on to the next generation''. Sel ...

A '''square number''', sometimes also called a '''perfect square''', is the result of an [[int ...exponentiation]]), usually pronounced as "''n'' squared". The name square number comes from the name of the shape; see [[#Properties|below]]. ...

...on]] within a hydrogen atom can only be [[integer]] multiples of a certain number. ...ful because it explains the cause of [[light]]. Bohr agreed with classical theory that light has a wave-particle duality (meaning that it is made of both [[e ...

The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one ...\mathbb{R}</math> is a function that maps each event to a nonnegative real number, satisfying ...

:''This article is about the basic notions. For advanced topics, see [[Group theory]].'' The number of elements in a group is called the group's ''order''. ...

...the positive [[divisor]]s of the number (except itself), the result is the number itself. 6 is the first perfect number. Its divisors (other than the number itself: 6) are 1, 2, and 3 and 1 + 2 + 3 equals 6. Other perfect numbers in ...

...ow that a [[mathematics|math]] [[theorem]] is true. One must show that the theory is true in all cases. ...lways true, precisely because it's true for whatever comes after any given number. ...

...th> players rolls a (fair) die. For any pair of players who throw the same number, the group scores <math>1</math> point. Find the mean and variance of the t == Problem 3 (Probability meets graph theory) == ...

Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can We consider a typical extremal problem for graphs: the largest possible number of edges of '''triangle-free''' graphs, i.e. graphs contains no <math>K_3</ ...

Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can We consider a typical extremal problem for graphs: the largest possible number of edges of '''triangle-free''' graphs, i.e. graphs contains no <math>K_3</ ...

Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can We consider a typical extremal problem for graphs: the largest possible number of edges of '''triangle-free''' graphs, i.e. graphs contains no <math>K_3</ ...

Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can We consider a typical extremal problem for graphs: the largest possible number of edges of '''triangle-free''' graphs, i.e. graphs contains no <math>K_3</ ...

Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can We consider a typical extremal problem for graphs: the largest possible number of edges of '''triangle-free''' graphs, i.e. graphs contains no <math>K_3</ ...

== Extremal Graph Theory == Extremal grap theory studies the problems like "how many edges that a graph <math>G</math> can ...

Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can We consider a typical extremal problem for graphs: the largest possible number of edges of '''triangle-free''' graphs, i.e. graphs contains no <math>K_3</ ...

The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one ...\mathbb{R}</math> is a function that maps each event to a nonnegative real number, satisfying ...

<li>[<strong>Random number of random variables</strong>] Let <math>\{X_n\}_{n \ge 1}</math> be identic ...rom the <strong>second</strong> round of tosses, and <math>Y</math> be the number of heads resulting from <strong>all</strong> tosses, which includes the fir ...

Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can We consider a typical extremal problem for graphs: the largest possible number of edges of '''triangle-free''' graphs, i.e. graphs contains no <math>K_3</ ...

Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can We consider a typical extremal problem for graphs: the largest possible number of edges of '''triangle-free''' graphs, i.e. graphs contains no <math>K_3</ ...

Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can We consider a typical extremal problem for graphs: the largest possible number of edges of '''triangle-free''' graphs, i.e. graphs contains no <math>K_3</ ...

...ntifier. Therefore, the sentence "every natural number has another natural number larger than it" is a quantified expression. Quantifiers and quantified expr : For each [[natural number]] ''n'', ''n'' · 2 = ''n'' + ''n''. ...

...of samples of various distributions. <ref>Gumbel, E.J. 1954. "Statistical theory of extreme values and some practical applications". ''Applied Mathematics S ...

The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one ...\mathbb{R}</math> is a function that maps each event to a nonnegative real number, satisfying ...

The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one ...\mathbb{R}</math> is a function that maps each event to a nonnegative real number, satisfying ...

The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one ...\mathbb{R}</math> is a function that maps each event to a nonnegative real number, satisfying ...

The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one ...\mathbb{R}</math> is a function that maps each event to a nonnegative real number, satisfying ...

The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one ...\mathbb{R}</math> is a function that maps each event to a nonnegative real number, satisfying ...

The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one ...\mathbb{R}</math> is a function that maps each event to a nonnegative real number, satisfying ...

The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one ...\mathbb{R}</math> is a function that maps each event to a nonnegative real number, satisfying ...

Note that in probability theory, the "mutual independence" is <font color="red">not</font> equivalent with ...<math>p</math>. Flipping the coin for n times, what is the expectation of number of HEADs? ...

=== Ramsey number === ...ath> satisfying the condition in the Ramsey theory is called the '''Ramsey number'''. ...

...olecular orbital]]s. This is used in the [[Hückel method]], [[ligand field theory]], and the [[Woodward–Hoffmann rules]]. Another idea on a larger scale is t ...ic point groups was published in 1936 by Rosenthal and Murphy.<ref>''Group Theory and the Vibrations of Polyatomic Molecules'' Jenny E. Rosenthal and G. M. M ...

...ny equations and many variables, not just two or three. In many cases, the number of equations and variables in the system are the same. In some cases, ther ...ds on the properties of the field. In many cases there will be an infinite number of solutions. ...

* [<strong>Random number of random variables</strong>] Let <math>\{X_n\}_{n \ge 1}</math> be identic ...rom the <strong>second</strong> round of tosses, and <math>Y</math> be the number of heads resulting from <strong>all</strong> tosses, which includes the fir ...

*[[set theory]] *[[Banach space]] theory ...

...be found by mathematically manipulating the wave function to return [[real number|real]] values relating to physical properties such as [[position]], [[momen The wave function can be in a number of different states at once, and so a particle may have many different posi ...

The permutation can be equivalently described as a composition of a number of '''cycles'''. For example, in the above permutation, we have two cycles: ...<math>G</math> be a permutation group acting on a set <math>X</math>. The number of orbits, denoted <math>|X/G|</math>, is ...