Why did a note scribbled in a book's margin take 358 years to prove?

Because the "truly marvelous proof" Pierre de Fermat claimed around 1637 probably never existed, and finding a real one took more than three centuries and whole new branches of mathematics.

The statement is simple. You know that 3² + 4² = 5²: squares can add up to a square in infinitely many ways. Fermat wrote, in the margin of his copy of Diophantus's Arithmetica, that the same never works for cubes or any higher power. There are no whole numbers with a³ + b³ = c³, or a⁴ + b⁴ = c⁴, and so on. He added that he had a marvelous proof, but the margin was too narrow to contain it.

Nobody could find it. In two centuries, mathematicians proved it only for powers 3, 5 and 7. By 1993, computers had checked every prime power below four million, but checking is not proving.

The breakthrough came from an unexpected link, discovered in the 1980s, to a deep problem about shapes called elliptic curves. Andrew Wiles, who had dreamed of proving Fermat's claim since finding a library book about it at age 10, worked on it in secret for over six years. After a nerve-racking flaw and another year of repairs, the proof was published in 1995, 358 years after Fermat's note.