• Takashi Ono
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
$$\mathop X\nolimits^2 + \mathop Y\nolimits^2 = \mathop Z\nolimits^2 $$
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.


Galois Group Integral Solution Diophantine Equation Great Common Divisor Prime Decomposition 
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    T. L. Heath, Diophantus of Alexandria, Dover, New York (1964).zbMATHGoogle Scholar
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    L. J. Mordell, Diophantine Equations, Academic Press, London-New York (1969).zbMATHGoogle Scholar
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    A. Weil, Number Theory: An Approach through History; from Hammurapi to Legendre Birkhäuser, Boston-Basel-Stuttgart (1983).Google Scholar

Copyright information

© Takashi Ono 1994

Authors and Affiliations

  • Takashi Ono
    • 1
  1. 1.The Johns Hopkins UniversityBaltimoreUSA

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