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Riker and Picard discuss Fermat’s Last Theorem in “The Royale.” Captain Picard: ‎”For 800 years people have been trying to solve it (Fermat’s Last Theorem)…”

Andrew Wiles did solve it after more than 350 years after Fermat first mentioned the proposition in the margin of his Arithmetica in 1637. Wiles proved FLT in 1994 and it was published in 1995. Apparently, the writers of ST TNG thought this theorem was completely inaccessible.

Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in the proof, by Andrew Wiles, of Fermat’s Last Theorem. More concretely, an elliptic curve is the set of zeros of a cubic polynomial in two variables. Where $ax^{3}+bx^{2}y+cxy^{2}+dy^{3}+ex^{2}+fxy+gy^{2}+hx+iy+j=0$ is the equation of a general cubic polynomial. A famous example being

$\displaystyle x^{3}+y^{3}=1$

or in homogeneous form,

$\displaystyle X^{3}+Y^{3}=Z^{3}$.

This is the first non-trivial case of Fermat’s Last Theorem.

A modular elliptic curve is an elliptic curve $E$ that admits a parametrization $X_{0}(N) \rightarrow E$ by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, and which could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is a modular form in disguise.

In 1985, starting with a fictitious solution to Fermat’s last theorem (the Frey curve), G. Frey showed that he could create an unusual elliptic curve which appeared not to be modular. If the curve were not modular, then this would show that if Fermat’s last theorem were false, then the Taniyama-Shimura conjecture would also be false. Furthermore, if the Taniyama-Shimura conjecture is true, then so is Fermat’s last theorem.

However, Frey did not actually prove that his curve was not modular. The conjecture that Frey’s curve was not modular came to be called the “epsilon conjecture,” and was quickly proved by Ribet (Ribet’s theorem) in 1986, establishing a very close link between two mathematical structures (the Taniyama-Shimura conjecture and Fermat’s last theorem) which appeared previously to be completely unrelated

By proving the semistable case of the conjecture, Andrew Wiles proved Fermat’s Last Theorem.

Some Elliptic curves:

This is a 45 minute documentary [Fermat’s Last Theorem (1996)] about Andrew Wiles, who proved Fermat’s Last Theorem in 1994. The theorem was first conjectured by Pierre de Fermat in 1637 and baffled mathematicians for more than 350 years! It states that no three positive integers $x$, $y$, and $z$ can satisfy the equation $x^n+y^n = z^n$ for $n>2$
The proof relied on elliptic curves and modular forms.