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- In [[number theory]], '''Wilson's Theorem''' states that if [[integer ]]<math>p > 1</math> , then <math>(p-1)! + 1</math> is divisible by <math>p ...inverse is unique, and each number is the inverse of its inverse. If one integer <math>a</math> is its own inverse, then4 KB (639 words) - 00:53, 2 February 2023
- ...ization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually <math>x</math> and <math>y</math> are variables and < *The integer <math>N</math> is positive. There are exactly <math>2005</math> ordered pai4 KB (682 words) - 12:13, 8 December 2024
- Its elementary algebraic formulation is often referred to as '''Cauchy's Inequality''' and states th ...ct vector spaces. Under this formulation, the elementary algebraic, linear algebraic, and calculus formulations are different cases of the general inequality.13 KB (2,048 words) - 14:28, 22 February 2024
- ...mathematics]] associated with studying the properties and identities of [[ integer]]s. ...pics such as [[Diophantine]] equations as well as some theorems concerning integer manipulation (like the [[Chicken McNugget Theorem ]]) are sometimes conside3 KB (399 words) - 22:08, 8 January 2024
- ...eans negative integers. The real numbers can also be divided between the [[algebraic number]]s and [[transcendental number]]s, although these two classes are b3 KB (496 words) - 22:22, 5 January 2022
- for any [[positive integer]]s <math>k</math> and <math>n</math>. Here, <math>\binom{n}{k}</math> is t ===Algebraic proof===12 KB (1,993 words) - 22:47, 27 October 2024
- ...number pi often shows up in problems in [[number theory]], particularly [[algebraic number theory]]. For example, many [[class number]] formulae involve pi. ...>x</math> is Liouvillian, then so is any rational multiple of any positive integer power of <math>x</math> (this is a simple exercise we leave to the reader),8 KB (1,469 words) - 20:11, 16 September 2022
- ...'' is a [[real number]] that cannot be expressed as the [[ratio]] of two [[integer]]s. Equivalently, an irrational number, when expressed in [[decimal notati There are two types of irrational numbers: algebraic and transcendental.3 KB (368 words) - 18:26, 6 June 2015
- ...] that is not a [[Root (polynomials) | root]] of any [[polynomial]] with [[integer|integral]] [[coefficient]]s. Many famous [[constant]]s such as [[pi | <mat * [[Algebraic number]]568 bytes (72 words) - 10:52, 9 December 2007
- ...erefore, contains at least one integer. Choosing <math>p</math> to be that integer, we immediately get the result. Let <math>n\ge 1</math> be any integer. Then there exists a rational number <math>\frac pq</math> such that <math>7 KB (1,290 words) - 11:18, 30 May 2019
- === Algebraic Proof === Now, since there are infinitely many real numbers but only finitely many integer pairs <math>(i,j)</math> with <math>1 \le i \le j \le d</math>, it follows5 KB (832 words) - 13:22, 11 January 2024
- <math>n</math> a postive integer. These are called the trivial zeros. * Koch, Helmut (trans. David Kramer), ''Number Theory: Algebraic Numbers and Functions.'' AMS 2000, ISBN 0-8218-2054-0.9 KB (1,547 words) - 02:04, 13 January 2021
- ...valence relation]] on the set of integers. This relation gives rise to an algebraic structure called '''the integers modulo <math>n</math>''' (usually known as Consider four integers <math>{a},{b},{c},{d}</math> and a positive integer <math>{n}</math> such that <math>a\equiv b\pmod {n}</math> and <math>c\equi14 KB (2,317 words) - 18:01, 29 October 2021
- .../math> let <math> S_n=\sum_{k=1}^{2^{n-1}}g(2k). </math> Find the greatest integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect ...g, but we can easily solve this problem by finding patterns, recursion and algebraic manipulations.10 KB (1,702 words) - 21:23, 25 July 2024
- ...ominators yields a polynomial with the same roots as <math>P(x)</math> and integer coefficients. The Rational Root Theorem can then be applied to the new poly ...he sign of every odd-degree coefficient, again minus some even nonnegative integer. Here roots are counted according to multiplicity, so double roots are coun8 KB (1,427 words) - 20:37, 13 March 2022
- ...ber of ordered quadruples <math>(x_1,x_2,x_3,x_4)</math> of positive odd [[integer]]s that satisfy <math>\sum_{i = 1}^4 x_i = 98.</math> Find <math>\frac n{1 Note that this is an algebraic bijection, we have simplified the problem and essentially removed the odd c5 KB (684 words) - 17:52, 19 June 2024
- ...<math>p</math> be a [[prime number|prime]], and let <math>a</math> be any integer. Then we can define the [[Legendre symbol]] ...ath> is a '''quadratic residue''' modulo <math>p</math> if there exists an integer <math>n</math> so that <math>n^2\equiv a\pmod p</math>.7 KB (1,182 words) - 15:46, 28 April 2016
- * [[Algebraic number]]s * [[Integer]]s458 bytes (62 words) - 12:35, 5 August 2017
- ..., [[ring]]s and [[field]]s (especially in the particular examples of the [[integer]]s, [[rational number]]s, [[real number]]s and [[complex number]]s). * [[Algebraic geometry]]3 KB (408 words) - 17:36, 11 December 2024
- An algebraic structure <math>\mathbb{S}</math> is said to have closure in a binary [[ope *The [[integer|integral]] number system <math>\mathbb{Z}</math> has closure in [[addition]1 KB (208 words) - 20:55, 20 August 2008