펠 방정식(Pell's equation) - 편집 역사

Pythagoras0: /* 메타데이터 */

2022-07-07T04:23:38Z

메타데이터

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2022-07-07T04:20:12Z

메타데이터: 새 문단

Pythagoras0: /* 노트 */ 새 문단

2022-07-07T04:20:11Z

노트: 새 문단

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2021-02-17T13:05:56Z

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-28T14:26:00Z

메타데이터: 새 문단

2020년 11월 13일 (금) 07:09에 Pythagoras0님의 편집

2020-11-13T07:09:07Z

2014년 10월 28일 (화) 21:50에 Pythagoras0님의 편집

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@@ 105번째 줄: / 105번째 줄: @@
 [[분류:초등정수론]]
 [[분류:디오판투스 방정식]]
 == 노트 ==

← 이전 판		2022년 7월 7일 (목) 04:20 판
145번째 줄:		145번째 줄:
	===소스===		===소스===
	<references />		<references />
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q853067 Q853067]
		+	===Spacy 패턴 목록===
		+	* [{'LOWER': 'pell'}, {'LOWER': "'s"}, {'LEMMA': 'equation'}]

← 이전 판		2022년 7월 7일 (목) 04:20 판
111번째 줄:		111번째 줄:
	===Spacy 패턴 목록===		===Spacy 패턴 목록===
	* [{'LEMMA': 'pell'}]		* [{'LEMMA': 'pell'}]
		+
		+	== 노트 ==
		+
		+	===말뭉치===
		+	# We can now see that is a nontrivial solution to pell's equation.<ref name="ref_f63d2248">[https://artofproblemsolving.com/wiki/index.php/Pell_equation Art of Problem Solving]</ref>
		+	# Therefore, such cannot exist and so the method of composition generates every possible solution to Pell's equation.<ref name="ref_f63d2248" />
		+	# According to Itô (1987), this equation can be solved completely using solutions to Pell's equation.<ref name="ref_cba6b0ff">[https://mathworld.wolfram.com/PellEquation.html Pell Equation -- from Wolfram MathWorld]</ref>
		+	# In this section we will concentrate on solutions to Pell's equation for the case where N = 1 and d > 0.<ref name="ref_fcc71f14">[https://www.math.uh.edu/~minru/web/pell4.html Section 13.4: "Pell's Equation"]</ref>
		+	# Hence we will look for solutions to Pell's equation in positive integers x and y. Suppose that we had such a solution.<ref name="ref_fcc71f14" />
		+	# Therefore a solution to Pell's equation will give us a good rational approximation to .<ref name="ref_fcc71f14" />
		+	# Those terms with second entry equal to 1 indicate solutions to Pell's equation.<ref name="ref_fcc71f14" />
		+	# Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions.<ref name="ref_e3e7603e">[https://en.wikipedia.org/wiki/Pell%27s_equation Pell's equation]</ref>
		+	# William Brouncker was the first European to solve Pell's equation.<ref name="ref_e3e7603e" />
		+	# Then the pair ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} solving Pell's equation and minimizing x satisfies x 1 = h i and y 1 = k i for some i. This pair is called the fundamental solution.<ref name="ref_e3e7603e" />
		+	# By this we mean simply: did Pell contribute at all to the study of Pell 's equation?<ref name="ref_0445b164">[https://mathshistory.st-andrews.ac.uk/HistTopics/Pell/ Pell's equation]</ref>
		+	# First let us say what Pell 's equation is.<ref name="ref_0445b164" />
		+	# In fact this method of composition allowed Brahmagupta to make a number of fundamental discoveries regarding Pell 's equation.<ref name="ref_0445b164" />
		+	# One property that he deduced was that ifsatisfies Pell 's equation so does.<ref name="ref_0445b164" />
		+	# It is well known that there exist an infinite number of integer solutions to the equation Dx^2+1=y^2 , known as Pell's equation .<ref name="ref_b3e84e09">[http://sweet.ua.pt/tos/pell.html Pell's equation]</ref>
		+	# Results concerning Pell's equation will be stated without proof.<ref name="ref_5e86d61f">[https://en.wiktionary.org/wiki/Pell%27s_equation Pell's equation]</ref>
		+	# This article gives the basic theory of Pell's equation x2 = 1 + D y2, where D ∈ ℕ is a parameter and x, y are integer variables.<ref name="ref_b703b1cb">[https://www.isa-afp.org/entries/Pell.html Pell's Equation]</ref>
		+	# The first part will discuss the history of Pell's Equation.<ref name="ref_316e6309">[https://repository.tcu.edu/handle/116099117/7234 Pell's Equation: History, Methods, and Number Theory]</ref>
		+	# Pell's Equation is an equation of the form x^2 - Dy^2 = 1, where x and y are variables in which integer solutions are sought and D is an integer.<ref name="ref_316e6309" />
		+	# We will then go into how Pell's Equation has even a longer history in India.<ref name="ref_316e6309" />
		+	# We will conclude this part by describing how Pell's Equation was known even in the time of Archimedes.<ref name="ref_316e6309" />
		+	# The name of Pell's equation arose from Leonhard Euler's mistakenly attributing its study to John Pell.<ref name="ref_f557b511">[https://www.definitions.net/definition/pell%27s+equation What does pell's equation mean?]</ref>
		+	# Bhaskara II in the 12th century and Narayana Pandit in the 14th century both found general solutions to Pell's equation and other quadratic indeterminate equations.<ref name="ref_f557b511" />
		+	# Task requirements find the smallest solution in positive integers to Pell's equation for n = {61, 109, 181, 277}.<ref name="ref_89ad567d">[https://rosettacode.org/wiki/Pell%27s_equation Pell's equation]</ref>
		+	# Thus, it got reduced to our Pell's Equation.<ref name="ref_388c2ce4">[http://reports.ias.ac.in/report/19831/pells-equation-and-rational-points-on-elliptic-curve Pell's equation and Rational points on elliptic curve]</ref>
		+	# When n is non-square, Pell's Equation has infinitely many solution pairs (X, Y), all of which can be generated by a single fundamental solution (X 1 , Y 1 ).<ref name="ref_8cd589b1">[https://www.had2know.org/academics/pell-equation-calculator.html Solutions to X^2 - nY^2 = 1]</ref>
		+	# To apply this method to solve Pell's equation, you simply compute the continued fraction of sqrt(n) and stop when the expansion begins to repeat.<ref name="ref_8cd589b1" />
		+	# Pell's equation is an important topic of algebraic number theory that involves quadratic forms and the structure of rings of integers in algebraic number fields.<ref name="ref_81d4534d">[https://www.goodreads.com/book/show/14361174-pell-s-equation Pell's Equation]</ref>
		+	===소스===
		+	<references />

@@ 106번째 줄: / 106번째 줄: @@
 [[분류:디오판투스 방정식]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q16881080 Q16881080]

← 이전 판		2020년 12월 28일 (월) 14:26 판
105번째 줄:		105번째 줄:
	[[분류:초등정수론]]		[[분류:초등정수론]]
	[[분류:디오판투스 방정식]]		[[분류:디오판투스 방정식]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q16881080 Q16881080]

@@ 27번째 줄: / 27번째 줄: @@
 이 정리를 이용하자.
-펠 방정식 <math>x_ {1}^2-dy_ {1}^2=1</math>의 정수해 $(x_1,y_1)$는
+펠 방정식 <math>x_ {1}^2-dy_ {1}^2=1</math>의 정수해 <math>(x_1,y_1)</math>는
 :<math>x_ {1}^2-dy_ {1}^2=(x_{1}+\sqrt{d}y_{1})(x_{1}-\sqrt{d}y_{1})=1</math>를 만족시키므로,
 :<math>|x_{1}-\sqrt{d}y_{1}|=\frac{1}{|x_{1}+\sqrt{d}y_{1}|}</math>

← 이전 판		2014년 10월 28일 (화) 21:50 판
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	[[분류:초등정수론]]		[[분류:초등정수론]]
		+	[[분류:디오판투스 방정식]]

@@ 76번째 줄: / 76번째 줄: @@
 * [[이차 수체(quadratic number fields) 의 정수론]]
 * [[연분수와 유리수 근사]]
 * [[체비셰프 다항식]]
 * [[루카스 수열]]

@@ 14번째 줄: / 14번째 줄: @@
 * <math>\sqrt{d}</math> 를 [[연분수와 유리수 근사|연분수]] 전개할때 얻어지는 convergents <math>{h_i}/{k_i}</math> 가 펠 방정식의 해가 되는 <math>x=h_i, y=k_i</math> 를 찾을 수 있으며, 이 때  <math>x</math>값을 가장 작게 하는 해를 fundamental solution 이라 한다.
-(정리)
+;정리
 펠 방정식의 해는 연분수 전개의 convergents 중에서 찾을 수 있다.
-(증명)
+;증명
 [[연분수와 유리수 근사]] 에서 펠 방정식에 관련한 중요한 정리는 다음과 같다
@@ 69번째 줄: / 69번째 줄: @@
 * [[수학사 연표]]
@@ 100번째 줄: / 92번째 줄: @@
 ==관련논문==
 * Lehmer, D. H. 1928. On the Multiple Solutions of the Pell Equation. The Annals of Mathematics 30, no. 1/4. Second Series (January 1): 66-72. doi:[http://dx.doi.org/10.2307/1968268 10.2307/1968268].
 [[분류:초등정수론]]

@@ 23번째 줄: / 23번째 줄: @@
 무리수 <math>\alpha</math>에 대하여, 유리수 <math>p/q</math>가 아래의 부등식을 만족시키는 경우,  <math>p/q</math>는 무리수 <math>\alpha</math>의 단순연분수 전개의 convergents 중의 하나이다
+:<math>|\alpha-\frac{p}{q}|<\frac{1}{2{q^2}}</math>
 이 정리를 이용하자.
-펠 방정식의 정수해 <math>x_ {1}^2-dy_ {1}^2=1</math> 는  <math>x_ {1}^2-dy_ {1}^2=(x_{1}+\sqrt{d}y_{1})(x_{1}-\sqrt{d}y_{1})=1</math>를 만족시키므로,
+펠 방정식 <math>x_ {1}^2-dy_ {1}^2=1</math>의 정수해 $(x_1,y_1)$는
+:<math>x_ {1}^2-dy_ {1}^2=(x_{1}+\sqrt{d}y_{1})(x_{1}-\sqrt{d}y_{1})=1</math>를 만족시키므로,
-<math>|x_{1}-\sqrt{d}y_{1}|=\frac{1}{|x_{1}+\sqrt{d}y_{1}|}</math>
+:<math>|x_{1}-\sqrt{d}y_{1}|=\frac{1}{|x_{1}+\sqrt{d}y_{1}|}</math>
+:<math>|\sqrt{d}-\frac{x_{1}}{y_{1}}|=\frac{1}{|x_{1}+\sqrt{d}y_{1}||y_{1}|}<\frac{1}{\sqrt{d}y_ {1}^{2}}\leq \frac{1}{2y_ {1}^{2}}</math>
+따라서,  펠 방정식의 해는 연분수 전개의 convergents 중에서 찾을 수 있다. ■
-==d=7인 경우==
+* <math>\sqrt{7}</math>의 연분수 전개를 통한 유리수근사:<math>\frac{2}{1},\frac{3}{1},\frac{5}{2},\frac{8}{3},\frac{37}{14}\cdots</math>
 *  펠 방정식의 해 찾기:<math>2^2-d\cdot 1^2=-3</math>:<math>3^2-d\cdot 1^2=2</math>:<math>5^2-d\cdot 2^2=-3</math>:<math>8^2-d\cdot 3^2=1</math>:<math>37^2-d\cdot 14^2=-3</math>
 * 따라서 펠 방정식 <math>x^2-7y^2=1</math>의 fundamental solution 은 <math>(8,3)</math> 이된다
+===d=13===
 * fundamental solution <math>(x_ 1,y_ 1)</math> 가 <math>y_ 1>6</math> 를 만족시키는 가장 작은 d
@@ 59번째 줄: / 53번째 줄: @@
-==d=61==
+===d=61===
@@ 65번째 줄: / 59번째 줄: @@
-==d=109==
+===d=109===
+*  페르마의 문제
+* <math>158070671986249^2 -109\cdot15140424455100^2=1</math>
@@ 96번째 줄: / 86번째 줄: @@
 * [[체비셰프 다항식]]
 * [[루카스 수열]]
 ==매스매티카 파일 및 계산 리소스==
@@ 101번째 줄: / 92번째 줄: @@
 * https://docs.google.com/leaf?id=0B8XXo8Tve1cxNTU4ZmMyMmQtMjNkZi00YWIwLWIzM2ItNzNiNTQ2YTRkMWY1&sort=name&layout=list&num=50
 * [http://projecteuler.net/problem=66 Project Euler, Problem 66]
 ==사전 형태의 자료==
@@ 115번째 줄: / 101번째 줄: @@
 ==관련논문==
-* [http://arxiv.org/abs/math/0311306 Conics - a Poor Man's Elliptic Curves]Franz Lemmermeyer, arXiv:math/0311306v1<br>
+* [http://arxiv.org/abs/math/0311306 Conics - a Poor Man's Elliptic Curves]Franz Lemmermeyer, arXiv:math/0311306v1
-* [http://www.ams.org/notices/200202/fea-lenstra.pdf Solving the Pell Equation]H. W. Lenstra Jr. Notices of the AMS 49 (2002), 182\[Dash]92
+* [http://www.ams.org/notices/200202/fea-lenstra.pdf Solving the Pell Equation]H. W. Lenstra Jr. Notices of the AMS 49 (2002), 182-92
 * Lehmer, D. H. 1928. On the Multiple Solutions of the Pell Equation. The Annals of Mathematics 30, no. 1/4. Second Series (January 1): 66-72. doi:[http://dx.doi.org/10.2307/1968268 10.2307/1968268].
-==블로그==
+[[분류:초등정수론]]