How Andrew Wiles Proved Fermat’s Last Theorem

For more than three centuries, Fermat’s Last Theorem remained one of the most famous unsolved problems in mathematics. Although the statement is easy to understand, no one could prove it using elementary methods.

Fermat’s Last Theorem states that the equation

$$x^n+y^n=z^n$$

has no non-zero integer solutions for any integer $n>\; 2\mathrm{.}$

The modern proof, completed by Andrew Wiles in the 1990s, did not study Fermat’s equation directly. Instead, it used an indirect argument and ideas from elliptic curves and modular forms.

An elliptic curve is a special type of curve defined by a mathematical equation, usually written as:

$$y^2 = x^3 + ax + b$$

where $a$ and $b$ are numbers chosen so that the curve has a smooth shape.

Example 1: $y^2=x^3-x+1$

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Example 2: $y^2 = x^3 + x + 1$

Despite the name, elliptic curves are not ellipses. The word “elliptic” comes from their historical connection to elliptic integrals.

The strategy begins by assuming that Fermat’s Last Theorem is false. If such a counterexample existed, Gerhard Frey showed that one could construct a special elliptic curve, now called the Frey curve. This curve would have very unusual properties and, in particular, it would be semistable.

Later, Ken Ribet proved that if this Frey curve existed, it could not be modular. On the other hand, Wiles proved that all semistable elliptic curves over the rational numbers are modular. These two conclusions contradict each other.

Because of this contradiction, the original assumption that Fermat’s Last Theorem is false must be wrong. Therefore, no such counterexample exists, and Fermat’s Last Theorem is true.