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The Fermat cubic surface
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation:
Therefore, in terms of the affine plane its equation is:
An integer solution to the Fermat equation would correspond to a nonzero rational number solution to the affine equation, and vice versa. But by Fermat's Last Theorem it is now known that (for n > 2) there are no nontrivial integer solutions to the Fermat equation; therefore, the Fermat curve has no nontrivial rational points.
The Fermat curve is non-singular and has genus:
This means genus 0 for the case n = 2 (a conic) and genus 1 only for n = 3 (an elliptic curve). The Jacobian variety of the Fermat curve has been studied in depth. It is isogenous to a product of simple abelian varieties with complex multiplication.
mathematics, the Fermatcurve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation: X n...
Pierre de Fermat (French: [pjɛʁ də fɛʁma]; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for...
completion of the curve in the projective plane and the points of the initial curve are those such that w is not zero. An example is the Fermatcurve un + vn =...
de Fermat, a French amateur mathematician. Fermat–Apollonius circle Fermat–Catalan conjecture Fermat cubic FermatcurveFermat–Euler theorem Fermat number...
In mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form: F n = 2 2 n +...
hyperelliptic curves, the Klein quartic curve, and the Fermatcurve xn + yn = zn when n is greater than three. Also projective plane curves in P 2 {\displaystyle...
dissertation on curved lines in comparison with straight lines). Building on his previous work with tangents, Fermat used the curve y = x 3 2 {\displaystyle...
superellipse is a plane algebraic curve of order pq. In particular, when a = b = 1 and n is an even integer, then it is a Fermatcurve of degree n. In that case...
^{3}z+\operatorname {sm} ^{3}z=1} , as real functions they parametrize the cubic Fermatcurve x 3 + y 3 = 1 {\displaystyle x^{3}+y^{3}=1} , just as the trigonometric...
seminal work on Fermat’s Last Theorem, Wiles set out to prove the modularity theorem for semistable elliptic curves, which implied Fermat’s Last Theorem...
elliptic curves. This curve was popularized in its application to Fermat’s Last Theorem where one investigates a (hypothetical) solution of Fermat's equation...
The Fermat primality test is a probabilistic test to determine whether a number is a probable prime. Fermat's little theorem states that if p is prime...
proof of Fermat's Last Theorem. They also find applications in elliptic curve cryptography (ECC) and integer factorization. An elliptic curve is not an...
image shows the standardized superelliptic Fermat squircle curve of the fourth degree: In a quartic Fermatcurve x 4 + y 4 = 1 {\displaystyle x^{4}+y^{4}=1}...
proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former...
Fermat's and Descartes' treatments is a matter of viewpoint: Fermat always started with an algebraic equation and then described the geometric curve that...
cubic plane curve defined from two diametrically opposite points of a circle. The curve was studied as early as 1653 by Pierre de Fermat, in 1703 by Guido...
polynomial x n + y n − 1 , {\displaystyle x^{n}+y^{n}-1,} which defines a Fermatcurve, is irreducible for every positive n. Over the field of reals, the degree...
from 1640; in 1659, Fermat stated to Huygens that he had proven the latter statement by the method of infinite descent. In 1657, Fermat posed the problem...
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