Diophantine Equations
A Diophantine Equation is characterized not by the shape of the equation but by the fact that one is interested only in solutions in integers (whole numbers) or rational numbers. Typically, it will have more than one variable. For example, xy = x + y + 2 can be looked at as a diophantine equation and it has only the solutions (x,y)=(2,4),(4,2),(0,-2),(-2,0) in integers. The equation 2x+4y = 1 is also a diophantine equation, and in this case it has no integral solutions, since the left hand side is even if x and y are integers and 1 is not even.
Historically, the first record of the study of Diophantine Equations is a Babylonian tablet, dating from before 1600 B.C., which lists integer solutions of the Pythagorean equation x2+y2=z2, which leads to right-angled triangles with integral sides. Diophantine Equations were also studied by the Greeks, in particular Diophantus of Alexandria (hence the name) around 200 AD because they preferred their problems to have integer or rational solutions. The great French mathematician Pierre de Fermat became interested in Diophantine Equations by reading a (then recent) translation of Diophantus's book Arithmetica into Latin, in the seventeenth century. He developed the subject extensively, creating the method of infinite descent which is still basic for the subject and by raising a number of interesting questions, some of which he said he could solve but did not divulge the solutions. This was meant as a challenge for his contemporaries such as B. Pascal and J. Wallis, but it not have the desired effect. The challenge was taken up only much later by other mathematicians, such as L. Euler who proved many of Fermat's assertions such as that any prime number of the form 4n+1 is a sum of two perfect squares, and Joseph-Louis Lagrange who proved that every integer is a sum of four perfect squares. Eventually, the only assertion made by Fermat left unproved was what became known as Fermat's last theorem which was only proved in 1993, by A. Wiles, almost 350 years after it was posed.
There is still much interest today in Diophantine equations, since our knowledge is very much incomplete. Study of Diophantine equations has been a driving force in the development of Mathematics. For instance, the theory of algebraic numbers was developed by E. Kummer partly as an attempt to prove Fermat's last theorem.
The simplest type of Diophantine equations are the linear ones. The basic case of two variables, that of equations of the type ax+by=c, can be solved by the Euclidean algorithm, as has been known since classical times. The so-called Pell equation (the name is due to a historical inaccuracy of Euler's) x2-dy2=1 was studied by the Indian mathematicians Brahmagupta in the sixth century AD and Bhaskara in the eleventh century AD, who had a method for finding the solutions. This method was rediscovered by Fermat and first published by Euler. There is a vast general theory for quadratic equations in many variables one of whose central results is the Hasse-Minkowski theorem that gives necessary and sufficient conditions for the solvability of homogeneous quadratic equations. Cubic equations in two variables fall into the theory of Elliptic curves which is a very developed theory but still an important topic of current research. A lot is known also about equations in two variables in higher degrees, specially since the German mathematician G. Faltings proved the Mordell conjecture in 1983, which ensures that equations that define the so-called curves of genus bigger than one have only finitely many rational solutions. Faltings received the Fields medal in 1986 for this work. For equations with more than three variables and degree at least three, very little is known. An important modern theme in the theory of Diophantine equations is the use of methods of Algebraic Geometry. One of the early proponents of this approach was A. Weil and the subject of Diophantine Geometry reached maturity with the work of A. Grothendieck who unified Number Theory and Algebraic Geometry with his theory of schemes.
Hilbert's tenth problem asked for an algorithm that would decide whether any given Diophantine equation had a solution or not. The Russian mathematician Yuri Matiyasevich proved in 1970, building on work of H. Putnam, M. Davis and J. Robinson, that such algorithm cannot possibly exist.
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