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Lagrange Number (Diophantine Equation)

Given a Fermat Difference Equation (a quadratic Diophantine Equation)

x^2-r^2 y^2=4

with $r$ a Quadratic Surd, assign to each solution $x\vert y$ the Lagrange number

z\equiv {\textstyle{1\over 2}}(x+yr).

The product and quotient of two Lagrange numbers are also Lagrange numbers. Furthermore, every Lagrange number is a Power of the smallest Lagrange number with an integral exponent.

See also Pell Equation


Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 94-95, 1965.

© 1996-9 Eric W. Weisstein