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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 ...