Part of the The University Series in Mathematics book series (USMA)
Finding a right triangle whose sides and hypotenuse have an integral length is equivalent to finding an ordered triple x, y, z of positive integers satisfying the equation
For example, 3, 4, 5; 4, 3, 5; 5, 12, 13; and 12, 5, 13 are solutions of (1.1). A solution such that the greatest common divisor of x, y, z is 1 is called a primitive solution. Since the polynomial \(\mathop X\nolimits^2 + \mathop Y\nolimits^2 = \mathop Z\nolimits^2 \) is homogeneous, every integral solution of (1.1) is a multiple of a primitive solution; hence it is enough to find all primitive solutions. Although the method of solving (1.1) is well-known, we review it here because the argument is very important and its central idea occurs over and over in this book.
$$\mathop X\nolimits^2 + \mathop Y\nolimits^2 = \mathop Z\nolimits^2 $$
KeywordsGalois Group Integral Solution Diophantine Equation Great Common Divisor Prime Decomposition
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© Takashi Ono 1994