Variations on a Theme of Euler pp 1-14 | Cite as

# Introduction

Chapter

## Abstract

Finding a right triangle whose sides and hypotenuse have an integral length is equivalent to finding an ordered triple
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*of positive integers satisfying the equation$$\mathop X\nolimits^2 + \mathop Y\nolimits^2 = \mathop Z\nolimits^2 $$

(1.1)

*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.## Keywords

Galois Group Integral Solution Diophantine Equation Great Common Divisor Prime Decomposition
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- 1.L. E. Dickson,
*History of the Theory of Numbers*, vol. 2, Chelsea, New York (1971).Google Scholar - 2.T. L. Heath,
*Diophantus of Alexandria*, Dover, New York (1964).zbMATHGoogle Scholar - 3.L. J. Mordell,
*Diophantine Equations*, Academic Press, London-New York (1969).zbMATHGoogle Scholar - 4.A. Weil,
*Number Theory: An Approach through History; from Hammurapi to Legendre*Birkhäuser, Boston-Basel-Stuttgart (1983).Google Scholar

## Copyright information

© Takashi Ono 1994