Why does Fermat’s last theorem exclude exponents greater than 2 but not the exponent 2?

Fermat’s Last Theorem does not exclude the exponent 2 because, for $n=2$, the equation $x^n+y^n=z^n$ (or $x^2+y^2=z^2$) has infinitely many solutions in natural numbers. These solutions are known as Pythagorean triples, such as 3, 4, and 5 ($3^2+4^2=5^2$), which satisfy the Pythagorean theorem.125

In contrast, Fermat’s Last Theorem specifically asserts that no such natural numbers exist for any exponent $n$ greater than 2. While Pierre de Fermat famously claimed to have a "truly remarkable proof" for this impossibility in the margin of his copy of Diophantus’s Arithmetica, the theorem remained one of the most famous unsolved problems in mathematics until it was finally proven by Andrew Wiles in 1995.132