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## Preview

Unable to display preview. Download preview PDF.

## 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).MATHGoogle Scholar - 3.L. J. Mordell,
*Diophantine Equations*, Academic Press, London-New York (1969).MATHGoogle 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