Diophantus of Alexandria c. He also made important advances in mathematical notation, and was one of the first mathematicians to introduce symbolism into algebra, using an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. He was perhaps the first to recognize fractions as numbers in their own right, allowing positive rational numbers for the coefficients and solutions of his equations. Diophantus applied himself to some quite complex algebraic problems, particularly what has since become known as Diophantine Analysis, which deals with finding integer solutions to kinds of problems that lead to equations in several unknowns. Diophantine equations Diophantine equations can be defined as polynomial equations with integer coefficients to which only integer solutions are sought. His general approach was to determine if a problem has infinitely many, or a finite number of solutions, or none at all.

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Coordinate Plane Note that any linear combination can be transformed into the linear equation , which is just the slope-intercept equation for a line. The solutions to the diophantine equation correspond to lattice points that lie on the line. For example, consider the equation or.

One solution is 0,1. Hence, the solutions to the equation may be written parametrically if we think of as a "starting point". Modular Arithmetic Sometimes, modular arithmetic can be used to prove that no solutions to a given Diophantine equation exist. Specifically, if we show that the equation in question is never true mod , for some integer , then we have shown that the equation is false.

However, this technique cannot be used to show that solutions to a Diophantine equation do exist. Induction Sometimes, when a few solutions have been found, induction can be used to find a family of solutions. Techniques such as infinite Descent can also show that no solutions to a particular equation exist, or that no solutions outside of a particular family exist.

General Solutions It is natural to ask whether there is a general solution for Diophantine equations, i. The answer, however, is no. In the s, Fermat, as he was working through a book on Diophantine Equations, wrote a comment in the margins to the effect of "I have a truly marvelous proof of this proposition which this margin is too narrow to contain.

After over years of failing to be proven, the theorem was finally proven by Andrew Wiles after he spent over 7 years working on the page proof, and another year fixing an error in the original proof. Problems Two farmers agree that pigs are worth dollars and that goats are worth dollars.

When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. For example, a dollar debt could be paid with two pigs, with one goat received in change. What is the amount of the smallest positive debt that can be resolved in this way?

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## DIOPHANTUS OF ALEXANDRIA

Biography[ edit ] Little is known about the life of Diophantus. He lived in Alexandria , Egypt , during the Roman era , probably from between AD and to or It was at first found that Diophantus lived between AD 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. Diophantus has variously been described by historians as either Greek , [2] [3] [4] non-Greek, [5] Hellenized Egyptian , [6] Hellenized Babylonian , [7] Jewish , or Chaldean. After consoling his fate by the science of numbers for four years, he ended his life.

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## Diophantus

Coordinate Plane Note that any linear combination can be transformed into the linear equation , which is just the slope-intercept equation for a line. The solutions to the diophantine equation correspond to lattice points that lie on the line. For example, consider the equation or. One solution is 0,1. Hence, the solutions to the equation may be written parametrically if we think of as a "starting point". Modular Arithmetic Sometimes, modular arithmetic can be used to prove that no solutions to a given Diophantine equation exist. Specifically, if we show that the equation in question is never true mod , for some integer , then we have shown that the equation is false.