Fermat's Last Theorem

Fermat's Last Theorem

Simon Singh

4.3(32063 readers)
The extraordinary story of the solving of a puzzle that has confounded mathematicians since the 17th century. The solution of Fermat’s Last Theorem is the most important mathematical development of the 20th century.
In 1963 a schoolboy browsing in his local library stumbled across the world’s greatest mathematical problem: Fermat’s Last Theorem, a puzzle that every child can understand but which has baffled mathematicians for over 300 years. Aged just ten, Andrew Wiles dreamed that he would crack it. Wiles’s lifelong obsession with a seemingly simple challenge set by a long-dead Frenchman is an emotional tale of sacrifice and extraordinary determination. In the end, Wiles was forced to work in secrecy and isolation for seven years, harnessing all the power of modern maths to achieve his childhood dream. Many before him had tried and failed, including a 18-century philanderer who was killed in a duel. An 18-century Frenchwoman made a major breakthrough in solving the riddle, but she had to attend maths lectures at the Ecole Polytechnique disguised as a man since women were forbidden entry to the school. A remarkable story of human endeavour and intellectual brilliance over three centuries, Fermat ‘s Last Theorem will fascinate both specialist and general readers.

Publisher

Fourth Estate

Publication Date

6/2/2005

ISBN

9781841157917

Pages

340

Categories

Mathematics

About the Author

Simon Singh

Simon Singh

Simon Lehna Singh, MBE is a British author who has specialised in writing about mathematical and scientific topics in an accessible manner. He is the maiden winner of the Lilavati Award.

His written works include Fermat's Last Theorem (in the United States titled Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem), The Code Book (about cryptography and its history), Big Bang (about the Big Bang theory and the origins of the universe) and Trick or Treatment? Alternative Medicine on Trial (about complementary and alternative medicine).

He has also produced documentaries and works for television to accompany his books, is a trustee of NESTA, the National Museum of Science and Industry and co-founded the Undergraduate Ambassadors Scheme.

Questions & Answers

Fermat's Last Theorem is a cornerstone in the history of mathematics, reflecting the evolution of mathematical thought and techniques. Its significance lies in its simplicity and enduring challenge. The theorem states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2. This simple statement has baffled mathematicians for over 350 years, leading to the development of numerous mathematical techniques and theories.

The theorem's evolution mirrors the progression of mathematical thought. It began with Pythagoras' theorem, which laid the groundwork for number theory. Fermat's theorem, though simple, required advanced techniques like modular forms, elliptic curves, and Galois representations, showcasing the evolution of mathematical tools. Euler's initial attempts and subsequent failures demonstrated the limits of traditional methods, while Andrew Wiles' proof in 1994, which relied on the Taniyama-Shimura conjecture, represented a monumental achievement, combining various mathematical fields.

The theorem's journey also reflects the collaborative nature of mathematics. It has inspired countless mathematicians to contribute to the field, leading to the development of new theories and techniques. Fermat's Last Theorem, therefore, is not just a single problem but a testament to the power of human ingenuity and the continuous evolution of mathematical thought.

Andrew Wiles's proof of Fermat's Last Theorem revolutionized number theory by unifying diverse mathematical areas. His work, which relied on the Taniyama-Shimura Conjecture, bridged elliptic equations and modular forms, creating a new understanding of these previously unrelated fields. This breakthrough not only resolved Fermat's Last Theorem but also laid the groundwork for the Langlands program, a broader initiative aimed at unifying various mathematical domains. Wiles's proof demonstrated the power of combining modern and traditional techniques, opening new avenues for solving other complex mathematical problems and contributing significantly to the pursuit of a unified mathematics.

Fermat's Last Theorem's solution involved several key mathematical concepts and techniques that evolved over time:

  1. Pythagorean Theorem: The theorem's foundation lies in Pythagoras' theorem, which states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This concept was expanded upon by Fermat, who posed the Last Theorem, which extends this idea to higher powers.

  2. Proof by Contradiction: Fermat's original proof for n=4 used this technique, which involves assuming a statement is true and then showing that this leads to a contradiction, thereby proving the original statement is false.

  3. Complex Numbers: Euler's work introduced complex numbers, which allowed for a more general approach to the problem. Euler used complex numbers to prove the theorem for n=3.

  4. Elliptic Curves: Andrew Wiles' proof relied heavily on elliptic curves, which are equations of the form y^2 = x^3 + ax + b. These curves have deep connections to number theory and were crucial in Wiles' proof.

  5. Taniyama-Shimura Conjecture: This conjecture, which Wiles proved, states that all elliptic curves over the rational numbers are modular. This implies Fermat's Last Theorem, as it shows that there are no solutions to the equation for n>2.

  6. Galois Representations: Wiles used Galois representations to connect elliptic curves to modular forms, which are functions that satisfy certain symmetry properties. This connection was essential in his proof.

Over time, these concepts and techniques were refined and expanded, leading to Wiles' groundbreaking proof in 1994. The evolution of these ideas demonstrates the interconnectedness of mathematics and the power of collaboration and perseverance in solving complex problems.

The personal stories of Fermat, Euler, Sophie Germain, and Wiles significantly contribute to the captivating narrative of Fermat's Last Theorem. Fermat's initial conjecture, based on a scribbled note, sparked a 350-year quest. Euler's breakthrough using the method of infinite descent and imaginary numbers laid the groundwork for future attempts. Sophie Germain, a pioneering female mathematician, revolutionized the study with her general approach, despite working under a pseudonym to overcome societal barriers. Andrew Wiles's decade-long struggle, involving the Taniyama-Shimura Conjecture, ultimately resolved the theorem. These mathematicians' dedication, perseverance, and the evolution of mathematical techniques intertwine to create a compelling story of human ingenuity and the beauty of mathematics.

Fermat's Last Theorem, with its profound implications, reshaped the nature of mathematical truth, beauty, and human understanding. Its simplicity and longevity as an unsolved problem underscore the beauty of mathematics, revealing its elegance and depth. The theorem's proof, involving complex and abstract concepts, demonstrated the power of mathematical abstraction and the importance of rigor in proof. It also highlighted the human quest for understanding, as mathematicians from various eras contributed to its solution, showcasing perseverance and creativity. The theorem's connection to other mathematical fields, like elliptic equations and modular forms, revealed the interconnectedness of mathematical knowledge, suggesting a grand unified theory of mathematics. Ultimately, Fermat's Last Theorem exemplifies the beauty of discovery, the pursuit of truth, and the endless potential of human intellect.

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