Diophantine Analysis And Fermats Last Theorem

The methods of Diophantus May cease to enchant us After a life spent trying to gear 'em To Fermat's Last Theorem.

An old chestnut, common in puzzle books of the late 19th century (when prices of farm animals were much lower than today), goes like this. A farmer spent $100 to buy 100 animals of three different kinds. Each cow cost $10, each pig $3 and each sheep 50 cents. Assuming that he bought at least one cow, one pig and one sheep, how many of each animal did the farmer buy?

At first glance this looks like a problem in elementary algebra, but the would-be solver quickly discovers that he has written a pair of simultaneous equations with three unknowns, each of which must have a value that is a positive integer. Finding integral solutions for equations is today called Diophantine analysis. In earlier centuries such analysis allowed integral fractions as values, but now it is usually restricted to whole numbers, including zero and negative integers. Of course in problems such as the one I have cited the values must be positive integers. Diophantine problems abound in puzzle literature. The well-known problem of the monkey and the coconuts, and the ancient task of finding right-angle triangles with integral sides, are among the classic instances of Diophantine problems.

The term "Diophantine" derives from Diophantus of Alex-^nrtria Hp was a nrominent Greek mathematician of his time, diophantine analysis and fermat's last theorem but to this day no one knows in what century he lived. Most authorities place him in the third century a.d. Nothing is known about him except some meager facts contained in a rhymed problem that appeared in a later collection of Greek puzzles. The verse has been quoted so often and its algebraic solution is so trivial, that I shall not repeat it here. If its facts are correct, we know that Diophantus had a son who died in his middle years and that Diophantus lived to the age of 84. About half of his major work, Arithmetka, has survived. Because many of its problems call for a solution in whole numbers, the term Diophantine became the name for such analysis. Diophantus made no attempt at a systematic theory, and almost nothing is known about Diophantine analysis by earlier mathematicians.

Today Diophantine analysis is a vast, complex branch of number theory with an enormous literature. There is a complete theory only for linear equations. No general method is known (it may not even exist) for solving equations with powers of 2 or higher. Even the simplest nonlinear Diophantine equation may be fantastically difficult to analyze. It may have no solution, an infinity of solutions or any finite number. Scores of such equations, so simple a child can understand them, have resisted all attempts either to find a solution or to prove none is possible.

The simplest nontrivial Diophantine equation has the linear form ax + by = c, where x and y are two unknowns and a, b and c are given integers. Let us see how such an equation underlies the puzzle in the opening paragraph. Letting x be the number of cows, y the number of pigs and z the number of sheep, we can write two equations:

To get rid of the fraction, multiply the first equation by 2. From this result, 20x 4- 631 + z = 200, subtract the second equation. This eliminates z, leaving 19x + 5;y = 100. How do we find integral values for * and y? There are many ways, but I shall ; give only an old algorithm that utilizes continued fractions and that applies to all equations of this form.

Put the term with the smallest coefficient on the left: 5y = 100-19*. Dividing both sides by 5 gives ^ = (100- 19x)/5. We next divide 100 and 19* by 5, putting the remainders (if any) over 5 to form a terminal fraction. In this way the equation is transformed to y = 20 — 3* - 4x/5.

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