Course outline
Module 1
Module 1: Introduction to Fermat's Last Theorem
Introduction to Fermat's Last Theorem
Knowledge check
Q1. Who proposed Fermat's Last Theorem?
- A. Leonhard Euler
- B. Pierre de Fermat ✓
- C. Andrew Wiles
- D. Carl Friedrich Gauss
Why: Pierre de Fermat proposed the theorem in the 17th century.
Q2. In which year did Fermat write his famous marginal note?
- A. 1637 ✓
- B. 1642
- C. 1650
- D. 1665
Why: Fermat wrote his marginal note in 1637 in his copy of Diophantus' Arithmetica.
Q3. What was the subject of Fermat's Last Theorem?
- A. The sum of two squares
- B. The sum of two cubes
- C. The sum of two squares and a cube
- D. The sum of two positive integers raised to a power greater than 2 ✓
Why: Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.
Q4. Who finally proved Fermat's Last Theorem?
- A. Leonhard Euler
- B. Carl Friedrich Gauss
- C. Andrew Wiles ✓
- D. Gerd Faltings
Why: Andrew Wiles proved Fermat's Last Theorem in 1994 after years of research.
Q5. What was the title of the book written by Simon Singh about Fermat's Last Theorem?
- A. The Code Book
- B. Fermat's Enigma ✓
- C. The Riemann Hypothesis
- D. The Music of the Primes
Why: Simon Singh wrote a book titled 'Fermat's Enigma' about the history and proof of Fermat's Last Theorem.
Module 2
Module 2: The Life and Work of Pierre de Fermat
The Life and Work of Pierre de Fermat
Knowledge check
Q1. What was Pierre de Fermat's primary profession?
- A. Mathematician
- B. Lawyer ✓
- C. Philosopher
- D. Physicist
Why: Fermat was a lawyer by profession and a mathematician as an amateur.
Q2. In which century was Pierre de Fermat born?
- A. 15th
- B. 16th ✓
- C. 17th
- D. 18th
Why: Fermat was born in the 16th century, specifically in 1601.
Q3. What is Fermat's Last Theorem commonly associated with?
- A. Algebra
- B. Number Theory ✓
- C. Geometry
- D. Calculus
Why: Fermat's Last Theorem is a famous problem in number theory.
Q4. What did Fermat write in the margin of his book regarding his theorem?
- A. I have found a truly remarkable proof of this theorem.
- B. I have discovered a proof that is too large to fit in the margin. ✓
- C. This theorem is incorrect.
- D. I have no proof for this theorem.
Why: Fermat famously wrote that he had a proof that was too large to fit in the margin.
Q5. Who finally proved Fermat's Last Theorem?
- A. Leonhard Euler
- B. Andrew Wiles ✓
- C. Carl Friedrich Gauss
- D. René Descartes
Why: Andrew Wiles proved Fermat's Last Theorem in 1994 after years of research.
Module 3
Module 3: The History of the Theorem
The History of the Theorem
Knowledge check
Q1. Who was the mathematician that first conjectured Fermat's Last Theorem?
- A. Leonhard Euler
- B. Pierre de Fermat ✓
- C. Andrew Wiles
- D. Carl Friedrich Gauss
Why: Pierre de Fermat was the mathematician who first conjectured Fermat's Last Theorem in the 17th century.
Q2. In which year did Fermat write his famous marginal note in his book?
- A. 1637 ✓
- B. 1642
- C. 1650
- D. 1665
Why: Fermat wrote his famous marginal note in 1637 in his copy of Diophantus' Arithmetica.
Q3. Which mathematician proved the case for Fermat's Last Theorem when the exponent is 4?
- A. Leonhard Euler ✓
- B. Sophie Germain
- C. Andrew Wiles
- D. Fermat himself
Why: Leonhard Euler proved the case for Fermat's Last Theorem when the exponent is 4 in the 18th century.
Q4. Which mathematician made significant progress on Fermat's Last Theorem in the early 19th century?
- A. Carl Friedrich Gauss
- B. Sophie Germain ✓
- C. Leonhard Euler
- D. Pierre de Fermat
Why: Sophie Germain made significant progress on Fermat's Last Theorem in the early 19th century by proving the theorem for a large class of exponents.
Q5. Which mathematician finally proved Fermat's Last Theorem in the 1990s?
- A. Carl Friedrich Gauss
- B. Andrew Wiles ✓
- C. Sophie Germain
- D. Leonhard Euler
Why: Andrew Wiles finally proved Fermat's Last Theorem in the 1990s after years of dedicated research.
Module 4
Module 4: Elliptic Curves and Modular Forms
Elliptic Curves and Modular Forms
Knowledge check
Q1. What is an elliptic curve typically defined as?
- A. A quadratic curve
- B. A cubic curve ✓
- C. A quartic curve
- D. A linear curve
Why: Elliptic curves are defined by cubic equations in two variables.
Q2. Which mathematician is closely associated with the development of modular forms?
- A. Leonhard Euler
- B. André Weil ✓
- C. Carl Friedrich Gauss
- D. Bernhard Riemann
Why: André Weil made significant contributions to the theory of modular forms.
Q3. What is a key property of elliptic curves used in number theory?
- A. They are always symmetric about the x-axis
- B. They have a group structure ✓
- C. They are defined over the real numbers only
- D. They are always rational curves
Why: Elliptic curves have a group structure, which is essential in number theory applications.
Q4. What is the modularity theorem also known as?
- A. Fermat's Last Theorem
- B. Taniyama-Shimura conjecture ✓
- C. Mordell's conjecture
- D. Euler's theorem
Why: The modularity theorem was originally conjectured by Taniyama and Shimura.
Q5. What is a modular form primarily associated with?
- A. Algebraic geometry
- B. Number theory ✓
- C. Differential equations
- D. Complex analysis
Why: Modular forms are deeply connected to number theory and have applications in various mathematical fields.
Module 5
Module 5: Wiles' Proof and Its Impact
Wiles' Proof and Its Impact
Module 6
Module 6: Mathematical Collaboration and Persistence
Mathematical Collaboration and Persistence
Module 7
Module 7: The Legacy of Fermat's Last Theorem
The Legacy of Fermat's Last Theorem
Knowledge check
Q1. What was the significance of Fermat's Last Theorem in the development of modern mathematics?
- A. It had no impact on modern mathematics.
- B. It inspired the development of new mathematical techniques and theories. ✓
- C. It was solved using only basic algebra.
- D. It was proven by Fermat himself.
Why: Fermat's Last Theorem inspired the development of new mathematical techniques and theories, particularly in number theory and algebraic geometry.
Q2. Who was the first person to prove Fermat's Last Theorem?
- A. Pierre de Fermat
- B. Andrew Wiles ✓
- C. Leonhard Euler
- D. Carl Friedrich Gauss
Why: Andrew Wiles was the first person to prove Fermat's Last Theorem in 1994.
Q3. What did Fermat write in the margin of his book that became famous?
- A. He wrote that he had found a proof for the theorem. ✓
- B. He wrote that he had discovered a new theorem unrelated to his work.
- C. He wrote that he had lost his notes on the proof.
- D. He wrote that he had forgotten how to solve the problem.
Why: Fermat wrote that he had found a proof for the theorem, but there was not enough space in the margin to write it down.
Q4. Which mathematical field was most directly influenced by the quest to solve Fermat's Last Theorem?
- A. Geometry
- B. Algebraic Number Theory ✓
- C. Calculus
- D. Trigonometry
Why: Algebraic Number Theory was most directly influenced by the quest to solve Fermat's Last Theorem, as it played a central role in Andrew Wiles' proof.
Q5. What was the impact of solving Fermat's Last Theorem on the mathematical community?
- A. It had no impact on the mathematical community.
- B. It brought widespread recognition to Andrew Wiles and inspired future research. ✓
- C. It caused a decline in interest in number theory.
- D. It led to the abandonment of other mathematical problems.
Why: Solving Fermat's Last Theorem brought widespread recognition to Andrew Wiles and inspired future research in mathematics.