Worked solution: Wiles's modularity proof via the Taylor–Wiles method (1994)

Fermat's Last Theorem asserts that $x^n + y^n = z^n$ has no solution in nonzero integers once $n \ge 3$. The proof by contradiction begins with a purely elementary normalization: it suffices to rule out the exponents $n = 4$ and odd primes $p \ge 5$, with the solution taken pairwise coprime (Wiles 1995, p. 443).

Indeed, every integer $n > 2$ is divisible either by $4$ or by an odd prime $p$. Writing $n = 4m$ or $n = pm$, a putative solution $x^n + y^n = z^n$ rewrites as $(x^m)^4 + (y^m)^4 = (z^m)^4$ or $(x^m)^p + (y^m)^p = (z^m)^p$, so any counterexample transports to one of these exponents. Fermat settled $n = 4$ around 1660 by infinite descent — a hypothetical solution generates a strictly smaller one, contradicting the well-ordering of the positive integers — and Euler settled $p = 3$ by 1770 through descent in $\mathbb{Z}[\sqrt{-3}]$ (with a gap later repaired by Legendre). Coprimality costs nothing: any prime $q$ dividing two of $a, b, c$ divides the third, so dividing the triple through by $\gcd(a,b,c)^p$ produces a pairwise coprime solution of the same equation.

The surviving data — a prime $p \ge 5$ and pairwise coprime nonzero $a, b, c$ with $a^p + b^p = c^p$ — are exactly the input that Frey's construction in Step 2 requires: the condition $p \ge 5$ will be essential when Serre's irreducibility theorem is invoked for $\overline{\rho}_{E,p}$, and coprimality is what forces the discriminant computation of Step 3 into its rigid $2p$-th-power shape.