디오판투스 방정식

개요

• 다변수다항식의 정수해 또는 유리수해를 찾는 문제를 디오판투스 방정식이라 함
• 부정방정식으로도 불림

예

• \(x^n+y^n=z^n\)의 정수해에 대한 페르마의 마지막 정리가 유명한 예
• \(x^2-dy^2=\pm 1\) 형태의 펠 방정식은 초등정수론과 대수적수론에서 중요한 역할을 함
• \(x^3+y^3=1729\)의 정수해. 라마누잔과 1729 참조
• \(3x^3+4y^3+5z^3=0\) (http://www.math.lsu.edu/~verrill/teaching/math7280/selmer_example/selmer_example.pdf)
• 합동수 문제 (congruent number problem)
• 사각 피라미드 퍼즐
• 펠 방정식(Pell's equation)
• 피타고라스 쌍(Pythagorean triple)
• 라마누잔-나겔 방정식

역사

• 수학사 연표

메모

• 모델 추측 (Mordell conjecture)

관련된 항목들

• 페르마의 마지막 정리
• 타원곡선
• 대수적수론
• 라마누잔과 1729

수학용어번역

• diophantine - 대한수학회 수학용어집
• 대한수학회 수학용어한글화 게시판

사전 형태의 자료

• http://ko.wikipedia.org/wiki/디오판투스방정식
• http://en.wikipedia.org/wiki/Diophantine_equation
• http://en.wikipedia.org/wiki/Thue-Siegel-Roth_theorem
• 에릭 웨이스타인, MathWorld - 디오판투스 방정식
• http://mathworld.wolfram.com/DiophantineEquation.html

리뷰, 에세이, 강의노트

• Pannekoek, René. “Diophantus Revisited: On Rational Surfaces and K3 Surfaces in the Arithmetica.” arXiv:1509.06138 [math], September 21, 2015. http://arxiv.org/abs/1509.06138.
• Das, Pranabesh, and Amos Turchet. “Invitation to Integral and Rational Points on Curves and Surfaces.” arXiv:1407.7750 [math], July 29, 2014. http://arxiv.org/abs/1407.7750.
• Ono, Ken. “Ramanujan, Taxicabs, Birthdates, ZIP Codes, and Twists.” The American Mathematical Monthly 104, no. 10 (December 1997): 912. doi:10.2307/2974471.

관련도서

• Diophantine equations
  • Mordell, L. J. (1969)

노트

위키데이터

• ID : Q905896

말뭉치

1. Diophantine equation, equation involving only sums, products, and powers in which all the constants are integers and the only solutions of interest are integers.
2. Congruence methods provide a useful tool in determining the number of solutions to a Diophantine equation.
3. Linear Diophantine equation in two variables takes the form of \(ax+by=c,\) where \(x, y \in \mathbb{Z}\) and a, b, c are integer constants.
4. : Solve the Homogeneous linear Diophantine equation \(6x+9y=0, x, y \in \mathbb{Z}\).
5. Use the following steps to solve a non-homogeneous linear Diophantine equation.
6. Finding the solution or solutions to a Diophantine equation is closely tied to modular arithmetic and number theory.
7. A Diophantine equation in the form is known as a linear combination.
8. The solutions to the diophantine equation correspond to lattice points that lie on the line.
9. Sometimes, modular arithmetic can be used to prove that no solutions to a given Diophantine equation exist.
10. A linear Diophantine equation is an equation of the first-degree whose solutions are restricted to integers.
11. This Diophantine equation was first studied extensively by Indian mathematician Brahmagupta around the year 628.
12. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one.
13. The simplest linear Diophantine equation takes the form ax + by = c, where a, b and c are given integers.
14. A homogeneous Diophantine equation is a Diophantine equation that is defined by a homogeneous polynomial.
15. If a Diophantine equation has as an additional variable or variables occurring as exponents, it is an exponential Diophantine equation.
16. We call a “Diophantine equation” to an equation of the form, \(f(x_1, x_2, \ldots x_n) = 0\) where \(n \geq 2\) and \(x_1, x_2, \ldots x_n\) are integer variables.
17. Hilbert's 10th problem asked if an algorithm existed for determining whether an arbitrary Diophantine equation has a solution.
18. We exhibit a collection of (probably infinitely many) rational numbers k for which this Diophantine equation is satisfied.
19. Brindza, B.: On a diophantine equation connected with the Fermat equation.

메타데이터

위키데이터

• ID : Q905896

Spacy 패턴 목록

• [{'LOWER': 'diophantine'}, {'LEMMA': 'equation'}]
• [{'LOWER': 'diophantic'}, {'LEMMA': 'equation'}]