Antimorph

A number which can be represented both in the form ${x_0}^2-D{y_0}^2$ and in the form $D{x_1}^2-{y_1}^2$. This is only possible when the Pell Equation

$$x^2-Dy^2=-1$$

is solvable. Then

$$x^2-Dy^2 = -(x_0-D{y_0}^2)({x_n}^2-D{y_n}^2)$$

$$= D(x_0y_n-y_0x_n)^2-(x_0x_n-Dy_0y_n)^2.$$

See also Idoneal Number, Polymorph

References

Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematical Entertains. New York: Dover, 1964.