펠 방정식(Pell's equation)

개요

• \(x^2-dy^2=1\) (\(d\) 는 완전제곱수를 약수로 갖지 않는 1보다 큰 자연수)형태의 디오판투스 방정식
• 연분수 전개를 통하여 모든 해를 구할 수 있음
• 해의 집합은 군의 구조를 통하여 이해할 수 있음
• \(x^2-dy^2=\pm 1\) 의 자연수 해를 구하는 문제는 실수 이차 수체의 unit 을 구하는 문제와 같음

연분수 전개와 fundamental solution

• \(\sqrt{d}\) 를 연분수 전개할때 얻어지는 convergents \({h_i}/{k_i}\) 가 펠 방정식의 해가 되는 \(x=h_i, y=k_i\) 를 찾을 수 있으며, 이 때 \(x\)값을 가장 작게 하는 해를 fundamental solution 이라 한다.

정리

펠 방정식의 해는 연분수 전개의 convergents 중에서 찾을 수 있다.

증명

연분수와 유리수 근사 에서 펠 방정식에 관련한 중요한 정리는 다음과 같다

무리수 \(\alpha\)에 대하여, 유리수 \(p/q\)가 아래의 부등식을 만족시키는 경우, \(p/q\)는 무리수 \(\alpha\)의 단순연분수 전개의 convergents 중의 하나이다 \[|\alpha-\frac{p}{q}|<\frac{1}{2{q^2}}\]

이 정리를 이용하자.

펠 방정식 \(x_ {1}^2-dy_ {1}^2=1\)의 정수해 \((x_1,y_1)\)는 \[x_ {1}^2-dy_ {1}^2=(x_{1}+\sqrt{d}y_{1})(x_{1}-\sqrt{d}y_{1})=1\]를 만족시키므로, \[|x_{1}-\sqrt{d}y_{1}|=\frac{1}{|x_{1}+\sqrt{d}y_{1}|}\] \[|\sqrt{d}-\frac{x_{1}}{y_{1}}|=\frac{1}{|x_{1}+\sqrt{d}y_{1}||y_{1}|}<\frac{1}{\sqrt{d}y_ {1}^{2}}\leq \frac{1}{2y_ {1}^{2}}\]

따라서, 펠 방정식의 해는 연분수 전개의 convergents 중에서 찾을 수 있다. ■

예

d=7인 경우

• \(\sqrt{7}\)의 연분수 전개를 통한 유리수근사\[\frac{2}{1},\frac{3}{1},\frac{5}{2},\frac{8}{3},\frac{37}{14}\cdots\]
• 펠 방정식의 해 찾기\[2^2-d\cdot 1^2=-3\]\[3^2-d\cdot 1^2=2\]\[5^2-d\cdot 2^2=-3\]\[8^2-d\cdot 3^2=1\]\[37^2-d\cdot 14^2=-3\]
• 따라서 펠 방정식 \(x^2-7y^2=1\)의 fundamental solution 은 \((8,3)\) 이된다

d=13

• fundamental solution \((x_ 1,y_ 1)\) 가 \(y_ 1>6\) 를 만족시키는 가장 작은 d
• \(649^2-13\cdot180^2=1\)

d=109

• 페르마의 문제
• \(158070671986249^2 -109\cdot15140424455100^2=1\)

역사

• 수학사 연표

관련된 항목들

• 이차 수체(quadratic number fields) 의 정수론
• 연분수와 유리수 근사
• 2의 제곱근(루트 2, 피타고라스 상수)
• 체비셰프 다항식
• 루카스 수열

매스매티카 파일 및 계산 리소스

• https://docs.google.com/leaf?id=0B8XXo8Tve1cxNTU4ZmMyMmQtMjNkZi00YWIwLWIzM2ItNzNiNTQ2YTRkMWY1&sort=name&layout=list&num=50
• Project Euler, Problem 66

사전 형태의 자료

• http://ko.wikipedia.org/wiki/펠방정식
• http://en.wikipedia.org/wiki/Pell's_equation

리뷰, 에세이, 강의노트

• Lemmermeyer, Franz. 2003. “Conics - a Poor Man’s Elliptic Curves.” arXiv:math/0311306 (November 18). http://arxiv.org/abs/math/0311306.
• Solving the Pell EquationH. W. Lenstra Jr. Notices of the AMS 49 (2002), 182-92

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:10.2307/1968268.

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노트

말뭉치

1. We can now see that is a nontrivial solution to pell's equation.
2. Therefore, such cannot exist and so the method of composition generates every possible solution to Pell's equation.
3. According to Itô (1987), this equation can be solved completely using solutions to Pell's equation.
4. In this section we will concentrate on solutions to Pell's equation for the case where N = 1 and d > 0.
5. Hence we will look for solutions to Pell's equation in positive integers x and y. Suppose that we had such a solution.
6. Therefore a solution to Pell's equation will give us a good rational approximation to .
7. Those terms with second entry equal to 1 indicate solutions to Pell's equation.
8. Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions.
9. William Brouncker was the first European to solve Pell's equation.
10. 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.
11. By this we mean simply: did Pell contribute at all to the study of Pell 's equation?
12. First let us say what Pell 's equation is.
13. In fact this method of composition allowed Brahmagupta to make a number of fundamental discoveries regarding Pell 's equation.
14. One property that he deduced was that ifsatisfies Pell 's equation so does.
15. 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 .
16. Results concerning Pell's equation will be stated without proof.
17. 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.
18. The first part will discuss the history of Pell's Equation.
19. 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.
20. We will then go into how Pell's Equation has even a longer history in India.
21. We will conclude this part by describing how Pell's Equation was known even in the time of Archimedes.
22. The name of Pell's equation arose from Leonhard Euler's mistakenly attributing its study to John Pell.
23. 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.
24. Task requirements find the smallest solution in positive integers to Pell's equation for n = {61, 109, 181, 277}.
25. Thus, it got reduced to our Pell's Equation.
26. 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 ).
27. 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.
28. 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.

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