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

References

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