Introduction Here, in the midst of this sad and barren landscape of the Greek ac- complishments in arithmetic, suddenly springs up a man with youthful energy: Diophantus. Where does he come from, where does he go to? Who were his predecessors, who his successors? We do not know. It is all one big riddle. He lived in Alexandria. If a conjecture were permitted, I would say he was not Greek . . . if his writings were not in Greek, no-one would ever think that they were an outgrowth of Greek culture . . . . Hermann Hankel (1874, translated by N. Schappacher) Diophantine equations have a long history. More than two thousand years ago, Diophantus of Alexandria considered, among many others, the equations x2 + y2 = z2 , (∗) y(6 y) = x3 x , and y2 = x2 + x4 + x8 . In Diophantus’ book Arithmetica, we find the formula ( (p2 q2)λ, 2pqλ, (p2 + q2)λ ) (†) that generates infinitely many solutions of (∗). For the second and third of the equations mentioned, Diophantus gives particular solutions, namely (1/36, 1/216) and (1/2, 9/16), respectively. In general, a polynomial equation in several indeterminates, where solutions are sought in integers or rational numbers, is called a Diophantine equation in honour of Diophantus. Diophantus himself was interested in solutions in positive integers or positive rational numbers. Contrary to the point of view usually adopted today, he did not accept negative numbers. It is remarkable that algebro-geometric methods have often been fruitful in order to understand a Diophantine equation. For example, there is a simple geometric idea behind formula (†). Indeed, since the equation is homogeneous, it suffices to look for solutions of X2 + Y 2 = 1 in rationals. This equation defines the unit circle. For every t , there is the line “x = −ty + 1” going through the point (1, 0). An easy calculation 1
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