The Quest to Prove Fermat's Last Theorem: Leading Up to Andrew Wiles' Groundbreaking Achievement

Fermat's Last Theorem, a conjecture proposed by the 17th-century mathematician Pierre de Fermat, has been the subject of extensive investigation over the centuries. The theorem states that there are no three positive integers (a), (b), and (c) that satisfy the equation (a^n b^n c^n) for any integer value of (n geq 2). Despite numerous attempts, no complete proof or disproof was found until Andrew Wiles' landmark achievement in 1994.

No One Proved Fermat's Last Theorem Before Wiles

The tale of Fermat's Last Theorem is filled with partial successes and missed opportunities. Though Fermat himself claimed to have a proof that he couldn't fit in the margin, no evidence supports the existence of such a proof. Likewise, many attempts by mathematicians over the centuries were incomplete or contained flaws. Ernst Eduard Kummer, a prominent 19th-century number theorist, made significant progress with his development of the theory of ideal numbers, proving the theorem for a specific class of primes. However, his work contained a flaw that prevented him from publishing it.

A History of Efforts to Prove the Theorem

The history of attempts to prove Fermat's Last Theorem is a testament to the enduring appeal and complexity of the problem. Here are some notable contributions:

Early Efforts

Pierre de Fermat: Fermat claimed to have a proof for the case (n 4), although the details of his argument have been lost. Pierre de Fermat: He also claimed to have a proof for the case (n 3), although this too was not recorded in a manner that could be verified. Sophie Germain: She contributed proofs for certain cases of the theorem.

Special Cases and Partial Proofs

Leonhard Euler: Proved the theorem for the case (n 3). Ernst Eduard Kummer: Developed the theory of ideal numbers and proved the theorem for a large class of primes that are either of the form (4k 1) or (4k 3).

The Connection to Modular Forms and Elliptic Curves

In the 20th century, mathematicians made significant strides in understanding the connection between modular forms and elliptic curves, which eventually led to Andrew Wiles' proof. The Taniyama-Shimura-Weil conjecture, now a theorem, suggested that every rational elliptic curve is modular. Wiles' proof of Fermat's Last Theorem heavily relied on this connection.

Conclusion

In summary, while many mathematicians such as Ernst Eduard Kummer made substantial partial progress, no complete proof of Fermat's Last Theorem had been established before Andrew Wiles' groundbreaking work in 1994. Wiles' achievement was not only a testament to the power of modern mathematics but also a capstone to centuries of mathematical inquiry.