Fermat’s Last Theorem

Fermat’s Last Theorem states that:

There are no positive integers $x, y,$ and $z$ with $x^n + y^n = z^n$ if $n$ is an integer greater than 2.

For $n = 2$ there are many solutions:

( 3, 4, 5) ( 5, 12, 13) ( 7, 24, 25) ( 8, 15, 17)
( 9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85)
(16, 63, 65) (20, 21, 29) (28, 45, 53) (33, 56, 65)
(36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97) … the Pythagorean triples.

In the margin of his copy of the Arithmetica of Diophantus the French jurist Fermat wrote circa 1637 that for greater $n$ no such triples can be found; he added that he had a marvelous proof for this, which however the margin was too small to contain:

Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatum in duos ejusdem nominis fas est dividere; cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caparet.

Some of the greatest mathematical minds in history have applied their genius to this problem, none succeeded. Every other result which Fermat had announced in like manner had long ago been dealt with; only this one, the last, remained. Finally, 358 years after Fermat wrote his theorem, Andrew Wiles saw an application of elliptic curves and modular forms (20th century mathematical constructs, not known to earlier mathematicians) that could contribute to the proof.

Theorem:

Let $n, a, b, c, \in \mathbb{Z}$ with $n > 2$.
If $a^n + b^n = c^n$ then $abc = 0.$

Proof:

The proof follows a program formulated around 1985 by Frey and Serre[F,S]. By classical results of Fermat, Euler, Dirichlet, Legendre, and Lamé, we may assume that $n = p$, an odd prime ≥ 11. Suppose $a, b, c \in \mathbb{Z} , abc \neq 0,$ and $a^p + b^p = c^p$. Without loss of generality we may assume that $2|a$ and $b \equiv 1 \bmod 4$. Frey [F] observed that the elliptic curve $E : y^2 = x(x -a^p )(x + b^p)$ has the following ‘remarkable’ properties:
(1) $E$ is semistable with conductor $N_E = \prod_{{l}|abc} {l}$ ; and
(2) $\overline{\rho}_{E,p}$ is unramified outside $2p$ and is flat at $p$ .
By the modularity theorem of Wiles and Taylor-Wiles [W,T-W], there is an eigenform $f \in S_2 (\Gamma_0(N_E))$ such that $\rho_{f,p} = \rho_{E,p}.$ A theorem of Mazur implies that $E,p$ is irreducible, so Ribet’s theorem [R] produces a Hecke eigenform $g \in S_2 (\Gamma_0(2))$ such that $\rho_{g,p} \equiv \rho_{f,p} \bmod {\tt p}$ for some ${\tt p}|p.$ But $X_0(2)$ has genus zero, so $S_2 (\Gamma_0(2)) = 0$. This is a contradiction and Fermat’s Last Theorem follows. Q.E.D.

References:

[F] Frey, G.: Links between semistable elliptic curves and certain Diophantine equations. Ann. Univ. Sarav. 1 (1986), 1-40.
[R] Ribet, K.: On modular representations of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ arising from modular forms. Invent. Math. 100 (1990), 431-476.
[S] Serre, J.-P.: Sur les représentations modulaires de degré 2 de $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ . Duke Math. J. 54 (1987), 179-230.
[T-W] Taylor, R. L., Wiles, A.: Ring-theoretic properties of certain Hecke algebras. Annals of Math. 141 (1995), 553-572.
[W] Wiles, A.: Modular elliptic curves and Fermat’s Last Theorem. Annals of Math. 141 (1995), 443-551.

It doesn’t fit the margin,
But it does go on a shirt.

Genealogy of a Theorem:

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 ~300 BC Euclid of Alexandria ~250 AD Diophantus of Alexandria 1601-1665 Pierre de Fermat 1707-1783 Leonhard Euler 1736-1813 Joseph Louis Lagrange 1776-1831 Sophie Germain 1777-1855 Carl Friedrich Gauss 1789-1857 Augustin Louis Cauchy 1795-1870 Gabriel Lamé 1805-1870 Peter Gustav Lejeune Dirichlet 1809-1882 Joseph Liouville 1810-1893 Ernst Eduard Kummer 1882-1973 Harry Schultz Vandiver Gerhard Frey Kenneth A. Ribet Andrew J. Wiles

(hat-tips: van der Poorten; Darmon and Levesque)