I Tried to Understand Fermat’s Last Theorem
I first heard about Fermat’s Last Theorem(FLT) in college when I was a math undergrad. For some reason, I have been fascinated by intellectual “beauty”. Something which is beautiful hence it exists. I have tried to pursue Art like I learnt painting for a few years or Learnt violin for a decade. Yet I couldn’t fully enjoy it as a practioner I love going to galleries and listening to music though. I understand the technicalities of music and art but I think Pure math and Engineering truly ever caught my fascination.
As a budding mathematician, I used to aspire to have such mathematical maturity that I am able to understand FLT deeply.
Around five months ago I took up a challenge (due to some personal reasons) to understand the proof in coming six months.
The reality is messier than I planned.
I am no where near to what I had imagined. However, it was still a rewarding experience for me personally.
I’m working full-time as an engineer at an early stage startup, my brain is almost always obsessing about the product, strategy, tech-culture etc. So it was hard to extract late-evening, weekend mornings where I could focus on the proof. Still, I stuck with it. Around three months went into elliptic curves going through Silverman, because I wanted to read the FLT proof in its actual language, not some washed-out version.
At some point things starting making some sense. I’m not saying I’m an expert, but I can now follow modular forms and algebraic number theory without getting totally lost. FLT forced me to learn how to learn this stuff properly.
The unexpected side-quest was Algebraic Topology. The proof’s ecosystem touches cohomology and deformation ideas — stuff I’d heard of but never had a reason to care about. I followed a topology lecture series along the way (link in a footnote), and while it is still new topic for me but I can now visualize a parallelogram becoming a torus!
Even if I didn’t hit my six-month goal, I can read advanced math now! Not fluently, but the ideas don’t feel alien anymore. I open a paper on algebraic geometry or cohomology and I can follow(with a lot of effort) it. I could have never imagined that I would ever be able to dive so deep into math alongside my day job!
Where I actually stand with FLT: I am only able to understand the structure of it. A long way to go still.
The proof structure (as I understand it)
The Setup
- For \(p>2\)
- Assume, there is a nontrivial coprime integer solution: \(a^p + b^p = c^p,\qquad abc\neq 0,\ \gcd(a,b,c)=1.\)
The Frey Curve
- From such a solution, build the Frey curve: \(E_{a,b,p}:\quad y^2 = x(x-a^p)(x+b^p).\)
- This curve is semistable over \(\mathbb{Q}\) and has extremely constrained reduction behavior.
The Galois Representation
- The \(p\) -torsion points give a mod-$p$ Galois representation: \(\bar\rho_{E,p}: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(\mathbb{F}_p).\)
- This representation is unramified outside primes dividing \(2abc\) , and (for \(p>7\) ) is irreducible.
Modularity
- The Modularity Conjecture(Taniyama-Shimura) - elliptic curves over \(\mathbb{Q}\) correspond to weight-2 modular forms.
- Wiles proved this for semistable elliptic curves. So the Frey curve would have to be modular.
Ribet’s Theorem
- Ribet proved Serre’s “epsilon conjecture” in this setting: if the Frey curve were modular, its mod- \(p\) representation would have to come from a weight-2 newform of level 2.
The Contradiction
- But there are no weight-2 newforms of level 2 (the space \(S_2(\Gamma_0(2))\) is zero).
- So a modular Frey curve cannot exist.
- Yet Wiles forces it to be modular.
- Contradiction.
- Therefore the original Fermat equation has no solutions for \(p>2\)
The actual Wiles proof is much more involved than this. The modularity theorem is proved by comparing deformation spaces of Galois representations to Hecke algebras — the famous (R=T) theorem — and that route goes through deformation theory, Selmer groups, and serious abstact algebra that I can only drop-names for now.
What surprised me wasn’t just the math, but the approach. Before going through this I used to think naively that FLT would have a number thoery or maybe a Proof-by-induction.
Real understanding comes from letting things simmer and an ability to chew things and not gulp them!
As a programmer, what sticks with me is the engineering of the proof!
You don’t solve hard problems by attacking them directly. You build the right abstractions — elliptic curves, Galois representations, modular forms — until the original problem becomes tractable or provably impossible inside a deeper framework.
This reminds me of two of Dijkstra quotes -
“The purpose of abstracting is not to be vague, but to create a new semantic level in which one can be absolutely precise.”
and
“The effective exploitation of his powers of abstraction must be regarded as one of the most vital activities of a competent programmer.”
I have some understanding of the skeleton now, yet so far from its intricacies which I would like to master. But the journey has already done what it was supposed to.
It exposed me to beautiful math and gave me a method for going deeper.
Pure math still captivates my imagination and it will, always! And the commitment I made six months ago? Still here. I just know now it’s going to take longer than I thought.
Footnote: [1] The topology lecture series I followed: Algebraic Topology by Math at Andrews University