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Lamé's special quartic is the graph of the equation

\( x^4 + y^4 = r^4 \)

where r > 0.[1] It looks like a rounded square with "sides" of length 2r and centered on the origin. This curve is a squircle centered on the origin, and it is a special case of a super ellipse.[2]

Because of Pierre de Fermat's only surviving proof, that of the n = 4 case of Fermat's Last Theorem, if r is rational there is no non-trivial rational point (x, y) on this curve (that is, no point for which both x and y are non-zero).
References

Oakley, Cletus Odia (1958), Analytic Geometry Problems, College Outline Series 108, Barnes & Noble, p. 171.
Schwartzman, Steven (1994), The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English, MAA Spectrum, Mathematical Association of America, p. 212, ISBN 9780883855119.

Mathematics Encyclopedia

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