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单词 PolynomialAnalogonForFermatsLastTheorem
释义

polynomial analogon for Fermat’s last theorem


For polynomialsPlanetmathPlanetmath with complex coefficients, there is an analogon of Fermat’s last theorem.  It can be proven quite elementarily by using Mason’s theorem (1983), but the original proof (about in 1900) was based on methods of algebraic geometryMathworldPlanetmathPlanetmathPlanetmath.

Theorem.  For an integer n greater than 2, there exist no non-constant coprimeMathworldPlanetmath polynomials x(t), y(t),z(t) in the ring [t] satisfying

[x(t)]n+[y(t)]n=[z(t)]n.(1)

Remark.  For  n=2, the equation (1) is in e.g. as

(2t)2+(1-t2)2=(1+t2)2.

References

  • 1 Serge Lang: “Die abc-Vermutung”.  – Elemente der Mathematik 48 (1993).
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