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Antimorph

An antimorph is a number which can be represented both in the form x_0^2-Dy_0^2 and in the form Dx_1^2-y_1^2. This is only possible when the Pell equation

x^2-Dy^2=-1
(1)

is solvable. Then

x^2-Dy^2=-(x_0-Dy_0^2)(x_n^2-Dy_n^2)
(2)
=D(x_0y_n-y_0x_n)^2-(x_0x_n-Dy_0y_n)^2.
(3)

See also

Idoneal Number, Polymorph

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References

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

Referenced on Wolfram|Alpha

Antimorph

Cite this as:

Weisstein, Eric W. "Antimorph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Antimorph.html

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