Questions: Fermat's Last Theorem (Overview)

Question 1

Andrew Wiles proved Fermat's Last Theorem by directly extending Fermat's infinite descent argument to most exponents n > 2.

• T. True
• F. False

Answer: False

Wiles's proof took a completely different route from Fermat's own methods. Infinite descent works for specific cases (Fermat used it for n=4; Euler handled n=3), but could not be generalized to all exponents. Wiles instead proved a crucial case of the Taniyama-Shimura conjecture about elliptic curves — an approach from algebraic geometry entirely unrelated to infinite descent. FLT followed as a corollary, not as a direct conclusion of any descent argument.

Question 2

Gerhard Frey's key observation was that a solution to x^n + y^n = z^n for n > 2 would produce an elliptic curve with which property?

• A. It would be a modular elliptic curve, confirming Taniyama-Shimura
• B. It could not be a modular elliptic curve, contradicting Taniyama-Shimura
• C. It would have no rational points, making Fermat's equation trivially unsolvable
• D. It would have infinitely many rational points, mirroring Pythagorean triples

Answer: (B)

Frey observed that a hypothetical FLT solution (a, b, c) with a^n + b^n = c^n would define an elliptic curve with such bizarre properties that it could not be modular. Ken Ribet then proved rigorously that this 'Frey curve' would violate the Taniyama-Shimura conjecture. So Taniyama-Shimura ⟹ no Frey curve ⟹ no FLT solution. Wiles proved the relevant case of Taniyama-Shimura, completing the chain. The common misconception is thinking Frey confirmed Taniyama-Shimura; he did the opposite — he showed a FLT solution would *contradict* it.

Question 3

Fermat's equation x^n + y^n = z^n has no positive integer solutions for any n > 2.

• T. True
• F. False

Answer: True

This is precisely what Fermat's Last Theorem states, and what Wiles proved in 1995. The contrast with n = 2 is sharp: infinitely many Pythagorean triples satisfy x² + y² = z², but the structure of the equation for n ≥ 3 is fundamentally different. Fermat conjectured this in 1637; it remained unproved until Wiles's 358-year-later resolution.

Question 4

A student claims: 'Mathematicians eventually proved FLT by checking enough cases computationally — once every exponent up to a billion was verified, the theorem was accepted.' What is wrong with this claim?

• A. Nothing — computational verification is sufficient for mathematical proof
• B. FLT was never computationally verified for large exponents, so the claim is factually wrong
• C. Checking finitely many cases cannot prove a statement about all integers n > 2, and Wiles's proof was non-computational
• D. The claim is wrong because Fermat himself verified all cases up to n = 100

Answer: (C)

Mathematical proof must cover all cases without exception — no finite computation can establish a universal claim about infinitely many exponents. FLT was indeed verified computationally for many specific exponents, which built confidence but was not a proof. Wiles's proof was a formal mathematical argument, not a computation. This misconception confuses empirical evidence with deductive proof — a crucial distinction in mathematics.

Question 5

Why was proving the Taniyama-Shimura conjecture (that every elliptic curve over the rationals is modular) sufficient to prove Fermat's Last Theorem?

Model answer: Frey showed that a hypothetical FLT solution would define an elliptic curve with properties too bizarre to be modular. Ribet proved this rigorously: any such 'Frey curve' would violate Taniyama-Shimura. So if Taniyama-Shimura is true (all elliptic curves are modular), the Frey curve cannot exist, which means no FLT solution can exist.

The logical structure is a proof by contradiction: assume FLT is false → construct Frey curve → Ribet's theorem says the Frey curve is not modular → this contradicts Taniyama-Shimura → so FLT must be true. Wiles proved the needed case of Taniyama-Shimura, closing the loop. This illustrates how FLT was not solved in isolation but by connecting it to deep structure in algebraic geometry — a bridge between distant fields of mathematics.