Diophantus looked at 3 different types of quadratic equations. Generalized solution in which the sides of triangle oab form a rational triple if line cb has a rational gradient t. On the exceptional set in the problem of diophantus and davenport. Find three numbers such that when any two of them are added, the sum is one of three given numbers. The problems in book i of the arithmetica are determinate ie, having a unique solution or a. We know very little about diophantus life, but you can find some of it here. Is there an english translation of diophantuss arithmetica. Diophantus is quoted around 350ce by theon of alexandria, heath, 2 giving. We can use his method to find solutions to the ops case, a 1. Five years after his marriage, was born a son who died 4 years before his father, at 1 2 log on. Diophantuss arithmetica1 is a list of about 128 algebraic problems with so lutions. His writing, the arithmetica, originally in books six survive in greek, another four in medieval arabic translation, sets out hundreds of arithmetic problems with their solutions. Problem 30 to nd two numbers whose sum and product are given 20,96.
This book features a host of problems, the most significant of. Thanks to an admirer of his, who described his life by means of an algebraic riddle, we know at least something about his life. Diophantus of alexandria, arithmetica and diophantine equations. Introduction the works of the mathematician diophantus have often struck readers as idiosyncratic.
The construction of each problem in arithmetica follows this pattern. We know little about this greek mathematician from alexandria, called the father of algebra, except that he lived around 3rd century a. The dating of his activity to the middle of the third century derives exclusively from a letter of michael psellus eleventh century. Book x presumably greek book vi deals with rightangled triangles with rational sides and subject to various further conditions. Derive the necessary condition on a and b that ensures a rational solution. We know virtually nothing about the life of diophantus. The problems in book i of the arithmetica are determinate ie, having a. 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. Diophantus and pappus ca 300 represent a shortlived revival of greek mathematics in a society that did not value math as the greeks had done 500750 years earlier. God grabted him to be a boy for the sixth part of his life, and adding a twelfth part to this, he clothed his cheeks with down. Algebra can essentially be considered as doing computations similar to those of arithmetic but with nonnumerical mathematical objects. He preformed the given operations and arrived at 35x 2 5, which according to diophantus is not a solution since it is not rational.
I feel as if, however, the wikipedia page, which states this contains both indeterminate and determinate equations might be slightly misleading, because i never encountered a definitively determinate equation. Iv into two books, at least other 2 manuscripts divide book i into two books. Nov 23, 2018 diophantus arithmetica consists of books written in greek in ce the dates vary by years from 70ad to ad. The sentence stating the determination can be easily recognized as such, since it immediately follows the complete enunciation. Example from diophantuss book ii, problem 8 divide a given square number, say 16, into the sum of two squares. It was at first found that diophantus lived between ad 250350 by analysing the price of wine used in many of his mathematical texts and finding out the period during which wine was sold at that price.
His book arithmetica is a collection of problems giving numerical solutions of. The date of diophantus death is the date of his sons birth plus his sons life plus 4, so. An example of this is found in problem 16, book i of the arithmetica, and it reads. Diophantus exists in an intermediate stage between rhetorical. The sentence stating the determination can be easily recognized as such, since it immediately follows the complete enunciation of the problem, it is. Since there was no symbol of zero when the book was written so diophantus took all coefficients to be positive. Diophantus considered negative or irrational square root solutions useless, meaningless, and even absurd. Theres just an abstract from the books, mostly an abbreviated description of the problems and their solutions which doesnt seem to be a 1. The known works of diophantus thus provide us with an interval of 500. Some claim that diophantus should not be called the father of algebra since his work contained mainly solutions to exact problems with no general solutions proposed.
For example, book ii, problem 8, seeks to express a given. Forty two problems of first degree from diophantus arithmetica a thesis by. The sentence stating the determination can be easily recognized as such, since it. At the close of the introduction, diophantus speaks of the thirteen books into which he had divided the. Diophantus passed 16 of his life in childhood, 112 in youth and 17 more as a bachelor. Archimedes has 51780 v3 265153 which is pretty accurate, but he does not tell how he got it. Diophantus of alexandria arithmetica book i joseph. This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. Of particular note is problem 8, since it is to this problem which fermat appended his famous last theorem. For example, book ii, problem 8, seeks to express a given square number as the sum of two square numbers here read more.
Diophantus lived in alexandria in times of roman domination ca 250 a. Another type of problem which diophantus studies, this time in book iv, is to find powers between given limits. Diophantine equations i putnam practice october 27, 2004 in his book arithmetica diophantus discussed the problem of. Alexandrian algebra according to diophantus rutgers university. If we accept, and we might as well, that the same diophantus was indeed also the author of the treatise on polygonal numbers, then we obtain a lower bound because in this work there is a quote from hypsicles who lived around 150 bcsee tannery 189395, vol. Diophantus wrote a seminal series of books called the arithmetica. Problem 31 to nd two numbers whose sum and sum of squares are given 20,208. The following is problem 7 of the first book of arithmetica. He lit him the light of wedlock after a seventh part, and five years after his marriger he granted a. He lived in alexandria, egypt, during the roman era, probably from between ad 200 and 214 to 284 or 298. Algebra customizable word problem solvers age solution.
This work was reproduced from the original artifact. Mar 30, 2007 diophantus back to the cool math games. Problem find two square numbers such that the sum of the product of the two numbers with either number is also a square number. Long ago diophantus of alexandria 4 noted that the numbers 116, 3316, 6816, and 10516 all have the property that the product of any two increased by 1 is the square of a rational number. Alexa denise rated it did not like it jul 20, 2018. Lets say the sums of three are 20,22,24 and 27 respectively. Diophantuss book is for the truly dedicated scholars and hobbyists who may still be searching for a proof for f. Babylonians yes v 2 pretty much ignored by the classical greeks because of the difficulty with irrational numbers. However, until the 19th century, algebra consisted essentially of the theory of equations. Alexandrian algebra according to diophantus mathematics. Diophantus arithmetica consists of books written in greek in ce the dates vary by years from 70ad to ad.
Diophantus noted that the rational numbers 116, 3316, 174 and 10516 have the following property. This book features a host of problems, the most significant of which have come to be called diophantine equations. For simplicity, modern notation is used, but the method is due to diophantus. Problem 29 to nd a number which when multiplied by two given numbers 200,5 makes one a square and the other its side. If we are to consider only the advancement of algebraic notation, then he was truly the father of algebra. Forty two problems of first degree from diophantus arithmetica a thesis by tinka davis bachelor of science, so. Diophantus s book is for the truly dedicated scholars and hobbyists who may still be searching for a proof for f. I feel i am sufficiently knowledgeable about the properties of quadratic relations.
To find two numbers such that their difference and the difference of their cubes are equal to two given numbers. The problem of diophantus and davenport for gaussian integers. If we take a birds eye view of arithmetica 6, we see that book i consists. One of these poems relates to the life, and the age at death, of a thirdcentury mathematician named diophantus, who lived in or around alexandria, egypt but was probably of greek heritage. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra in fact, every proof must use.
The known works of diophantus thus provide us with an interval of 500 years for their composition, and this is about all that we may be certain of when it comes to dating diophantus of alexandria. Diophantuss main achievement was the arithmetica, a collection of arithmetical problems involving the solution of determinate and indeterminate equations. Diophantus life span problem diophantus youth lasted 16 of his life. It seems more like a book about diophantuss arithmetica, not the translation of the actual book. Assume that this square is 2x 4 where xis the unknown. And if diophantus states a necessary condition for dividing a number into two or three squares as in the previous case of v. Thanks to an admirer of his, who described his life by means of an algebraic riddle, we know at least something about. One of the most famous problems that diophantus treated was writing a square as the sum of two squares book ii, problem 8. It seems more like a book about diophantus s arithmetica, not the translation of the actual book. Pdf a problem of diophantus and dicksons conjecture. In it he introduced algebraic manipulations on equations including a symbol for one unknown probably following other authors in alexandria. By using the modern notation, let one of the required squares be x2. Find two numbers such that their sum and product are given. The following is a statement of arithmetica book ii, problem 28 and its solution.
To get my answer i used an excel worksheet and started at 10,001 and made a list of integers 9,996, 9991, 9986 etc. We may generalize diophantuss solution to solve the problem for any given square, which we will represent algebraically as a 2. The son lived exactly 1 2 as long as his father, and diophantus died. This riddle about diophantuss age when he died was carved on his tomb. The eighth problem of the second book of diophantuss arithmetica is to divide a square into a sum of two squares. The meaning of plasmatikon in diophantus arithmetica. Diophantus back to the cool math games we know little about this greek mathematician from alexandria, called the father of algebra, except that he lived around 3rd century a. Problem 8 illustrates how adroitly diophantus is able to handle this limitation. In other words, for the given numbers a and b, to find x and y such that x y a and x3 y3 b. A similar problem involves decomposing a given integer into the sum of three squares. The reason why there were three cases to diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers a, b, c to all be positive in each of the three cases above. My answer, which was correct, but not the lowest possible answer was 31,246 coconuts in the original pile. 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 be written as a sum of two cubes, fourth powers not as a sum of two fourth pow.
315 1023 51 1316 1355 771 818 230 14 1507 809 821 1304 1372 496 1403 333 824 1053 1507 978 445 459 1474 144 18 331 667 161 166 1367 1304 632 304 1593 1451 1270 1068 821 547 818 230 973 299 536 1194 1124