Abstract
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.
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References
L. E. Dickson, History of the Theory of Numbers, vol. 2, Chelsea, New York (1971).
T. L. Heath, Diophantus of Alexandria, Dover, New York (1964).
L. J. Mordell, Diophantine Equations, Academic Press, London-New York (1969).
A. Weil, Number Theory: An Approach through History; from Hammurapi to Legendre Birkhäuser, Boston-Basel-Stuttgart (1983).
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© 1994 Takashi Ono
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Ono, T. (1994). Introduction. In: Variations on a Theme of Euler. The University Series in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2326-7_1
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DOI: https://doi.org/10.1007/978-1-4757-2326-7_1
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-3241-9
Online ISBN: 978-1-4757-2326-7
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