To divide a given square into a sum of two squares. On the contrary, it is impossible to divide either a cube into two cubes. This problem became important when fermat, in his copy of diophantus arithmetica edited by bachet, noted that he had this wonderful proof that cubes cant. For example, book ii, problem 8, seeks to express a given square number as the sum of two square numbers here read more. Find two square numbers whose di erence is a given number, say 60. Just in the same manner as in diophantus the multiplication of two species. Abels step ii implies that r 5 is a 2 3 4 rational function of the roots. Intersection of the line cb and the circle gives a rational point x 0,y 0. The problems in book i of the arithmetica are determinate ie, having a unique solution or a. The problems one of the most famous problems that diophantus treated was writing a square as the sum of two squares book ii, problem 8. Diophantus of alexandria arithmetica book i joseph.
The eighth problem of the second book of diophantuss arithmetica is to divide a square into a sum of two squares. Mathematics in the time of the pharos free ebook download as pdf file. This book features a host of problems, the most significant of which have come to be called diophantine equations. Book x presumably greek book vi deals with rightangled triangles with rational sides and subject to various further conditions. Thus the problem has been reduced to a linear equation, which. Alexandrian algebra according to diophantus mathematics. Diophantus of alexandria, arithmetica and diophantine equations.
The first ten propositions of book ii can be easily interpreted in modern algebraic notation. Murty,esmonde problems in algebraic number theory summation. Diophantuss arithmetica1 is a list of about 128 algebraic problems with so lutions. Diophantus of alexandria had a great impact in the world of mathematics. Scribd is the worlds largest social reading and publishing site. It consists of a short paragraph from theons commentary on the first book of. If a problem leads to an equation in which certain terms are equal to terms of the same species. In it he introduced algebraic manipulations on equations including a symbol for one unknown probably following other authors in alexandria. This book tells the story of abels problem and his proof.