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Lamé's special quartic

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Lamé's special quartic with "radius" 1.

Lamé's special quartic, named after Gabriel Lamé, is the graph of the equation

x 4 + y 4 = r 4 {\displaystyle x^{4}+y^{4}=r^{4}}

where r > 0 {\displaystyle r>0} . It looks like a rounded square with "sides" of length 2 r {\displaystyle 2r} and centered on the origin. This curve is a squircle centered on the origin, and it is a special case of a superellipse.

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 rational numbers).

References

  1. Oakley, Cletus Odia (1958), Analytic Geometry Problems, College Outline Series, vol. 108, Barnes & Noble, p. 171.
  2. 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.


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