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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Pellian equation ： ウィキペディア英語版 Pell's equation Pell's equation (also called the Pell–Fermat equation) is any Diophantine equation of the form :$x^2-ny^2=1\backslash ,$ where ''n'' is a given positive nonsquare integer and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, the equation has the form of a hyperbola; solutions occur wherever the curve passes through a point whose ''x'' and ''y'' coordinates are both integers, such as the trivial solution with ''x'' = 1 and ''y'' = 0. Joseph Louis Lagrange proved that, as long as ''n'' is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of ''n'' by rational numbers of the form ''x/y''. This equation was first studied extensively in India, starting with Brahmagupta, who developed the ''chakravala'' method to solve Pell's equation and other quadratic indeterminate equations in his ''Brahma Sphuta Siddhanta'' in 628, about a thousand years before Pell's time. His ''Brahma Sphuta Siddhanta'' was translated into Arabic in 773 and was subsequently translated into Latin in 1126. 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. Solutions to specific examples of the Pell equation, such as the Pell numbers arising from the equation with ''n'' = 2, had been known for much longer, since the time of Pythagoras in Greece and to a similar date in India. The name of Pell's equation arose from Leonhard Euler's mistakenly attributing Lord Brouncker's solution of the equation to John Pell.〔Lettre IX. Euler à Goldbach, dated 10 August 1750 in: P. H. Fuss, ed., ''Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siècle'' … (Mathematical and physical correspondence of some famous geometers of the 18th century), vol. 1, (St. Petersburg, Russia: 1843), pp. 35-39 ; see especially (page 37. ) From page 37: ''"Pro hujusmodi quaestionibus solvendis excogitavit D. Pell Anglus peculiarem methodum in Wallisii operibus expositam."'' (For solving such questions, the Englishman Dr. Pell devised a singular method (is ) shown in Wallis' works.)〕 For a more detailed discussion of much of the material here, see Lenstra (2002) and Barbeau (2003). ==History== As early as 400 BC in India and Greece, mathematicians studied the numbers arising from the ''n'' = 2 case of Pell's equation, :$x^2\; -\; 2y^2=1$ and from the closely related equation :$x^2\; -\; 2y^2\; =\; -1$ because of the connection of these equations to the square root of two.〔 Indeed, if ''x'' and ''y'' are positive integers satisfying this equation, then ''x''/''y'' is an approximation of √2. The numbers ''x'' and ''y'' appearing in these approximations, called side and diameter numbers, were known to the Pythagoreans, and Proclus observed that in the opposite direction these numbers obeyed one of these two equations.〔 Similarly, Baudhayana discovered that ''x'' = 17, ''y'' = 12 and ''x'' = 577, ''y'' = 408 are two solutions to the Pell equation, and that 17/12 and 577/408 are very close approximations to the square root of two. Later, Archimedes approximated the square root of 3 by the rational number 1351/780. Although he did not explain his methods, this approximation may be obtained in the same way, as a solution to Pell's equation.〔.〕 Archimedes' cattle problem involves solving a Pellian equation, though it is unclear whether this problem is really due to Archimedes. Around AD 250, Diophantus considered the equation :$a^2\; x^2+c=y^2,$ where ''a'' and ''c'' are fixed numbers and ''x'' and ''y'' are the variables to be solved for. This equation is different in form from Pell's equation but equivalent to it. Diophantus solved the equation for (''a'',''c'') equal to (1,1), (1,−1), (1,12), and (3,9). Al-Karaji, a 10th-century Persian mathematician, worked on similar problems to Diophantus. In Indian mathematics, Brahmagupta discovered that :$(x\_1^2\; -\; Ny\_1^2)(x\_2^2\; -\; Ny\_2^2)\; =\; (x\_1x\_2\; +\; Ny\_1y\_2)^2\; -\; N(x\_1y\_2\; +\; x\_2y\_1)^2\; =\; (x\_1x\_2\; -\; Ny\_1y\_2)^2\; -\; N(x\_1y\_2\; -\; x\_2y\_1)^2$ (see Brahmagupta's identity). Using this, he was able to "compose" triples $(x\_1,\; y\_1,\; k\_1)$ and $(x\_2,\; y\_2,\; k\_2)$ that were solutions of $x^2\; -\; Ny^2\; =\; k$, to generate the new triple :$(x\_1x\_2\; +\; Ny\_1y\_2\; \backslash ,,\backslash ,\; x\_1y\_2\; +\; x\_2y\_1\; \backslash ,,\backslash ,\; k\_1k\_2)$ and $(x\_1x\_2\; -\; Ny\_1y\_2\; \backslash ,,\backslash ,\; x\_1y\_2\; -\; x\_2y\_1\; \backslash ,,\backslash ,\; k\_1k\_2).$ Not only did this give a way to generate infinitely many solutions to $x^2\; -\; Ny^2\; =\; 1$ starting with one solution, but also, by dividing such a composition by $k\_1k\_2$, integer or "nearly integer" solutions could often be obtained. For instance, for $N=92$, Brahmagupta composed the triple $(10,\; 1,\; 8)$ (since $10^2\; -\; 92(1^2)\; =\; 8$) with itself to get the new triple $(192,\; 20,\; 64)$. Dividing throughout by 64 gave the triple $(24,\; 5/2,\; 1)$, which when composed with itself gave the desired integer solution $(1151,\; 120,\; 1)$. Brahmagupta solved many Pell equations with this method; in particular he showed how to obtain solutions starting from an integer solution of $x^2\; -\; Ny^2\; =\; k$ for ''k'' = ±1, ±2, or ±4. The first general method for solving the Pell equation (for all ''N'') was given by Bhaskara II in 1150, extending the methods of Brahmagupta. Called the chakravala (cyclic) method, it starts by composing any triple $(a,b,k)$ (that is, one which satisfies $a^2\; -\; Nb^2\; =\; k$) with the trivial triple $(m,\; 1,\; m^2\; -\; N)$ to get the triple $(am\; +\; Nb,\; a+bm,\; k(m^2-N))$, which can be scaled down to :$\backslash left(\; \backslash frac\; \backslash ,,\backslash ,\; \backslash frac\; \backslash ,,\backslash ,\; \backslash frac\; \backslash right).$ When ''m'' is chosen so that ''(a+bm)/k'' is an integer, so are the other two numbers in the triple. Among such ''m'', the method chooses one that minimizes ''(m²-N)/k'', and repeats the process. This method always terminates with a solution (proved by Lagrange in 1768). Bhaskara used it to give the solution ''x''=1766319049, ''y''=226153980 to the notorious ''N'' = 61 case.〔 Several European mathematicians rediscovered how to solve Pell's equation in the 17th century, apparently unaware that it had been solved almost a thousand years earlier in India. Fermat found how to solve the equation and in a 1657 letter issued it as a challenge to English mathematicians. In a letter to Digby, Bernard Frénicle de Bessy said that Fermat found the smallest solution for ''N'' up to 150, and challenged John Wallis to solve the cases ''N'' = 151 or 313. Both Wallis and Lord Brouncker gave solutions to these problems, though Wallis suggests in a letter that the solution was due to Brouncker. Pell's connection with the equation is that he revised Thomas Branker's translation of Johann Rahn's 1659 book "Teutsche Algebra" into English, with a discussion of Brouncker's solution of the equation. Euler mistakenly thought that this solution was due to Pell, as a result of which he named the equation after Pell. The general theory of Pell's equation, based on continued fractions and algebraic manipulations with numbers of the form $P+Q\backslash sqrt,$ was developed by Lagrange in 1766–1769.〔"Solution d'un Problème d'Arithmétique", in (J.-A. Serret (Ed.), ''Oeuvres de Lagrange'', vol. 1, pp. 671–731, 1867. )〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pell's equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.