My notebook from 3 years ago: A tribute to how far I've come
Earlier, when I worked with congruences and large exponents, my approach was careful, almost anxious. I would recognize that a modulus was prime and immediately reach for Fermat’s Little Theorem. From there, I tried to force the exponent into the right shape. Reduce here, subtract there, check again. Every step felt fragile. Even when I arrived at the right answer, it didn’t feel inevitable. It felt like I had successfully navigated a maze without quite understanding its layout.
What I was missing then was not technique. It was orientation.
I was treating modular arithmetic as a collection of permitted manipulations rather than as a mathematical object in its own right. Reduction modulo felt like information loss, something dangerous that had to be justified repeatedly. I worried about whether I was allowed to cancel, whether something subtle was collapsing, whether I was cheating by making numbers smaller. Those worries were signals. They weren’t signs of weakness, but signs that I didn’t yet see the structure holding everything together.
In hindsight, I can see that I was working inside a group without realizing it. The multiplicative residues modulo a prime form a finite group. Exponents cycle because the group is finite. Fermat’s Little Theorem is not a trick. It is a shadow cast by that structure. But without that mental picture, every reduction felt like a local hack instead of a global consequence.
What’s changed now is subtle but decisive. When I see an expression like , I no longer see a huge number that needs to be tamed. I see a finite state space. I expect periodicity. I expect that the exponent will eventually stop mattering in the way it appears to. The computation feels calm because I’m no longer fighting the size of the numbers. I’m working inside an object whose behavior I trust.
The same contrast shows up even more starkly with roots modulo a prime. Earlier, I approached them with the intuition I had from real numbers, half-expecting some notion of approximation or gradual improvement. That intuition doesn’t belong here. A square root modulo is not something you approach. It either exists or it doesn’t. The question is algebraic, not analytic. Once I started seeing roots as questions about group structure rather than numerical magnitude, the problem snapped into focus.
What’s striking is that my difficulty back then was not about understanding statements or following proofs. It was about premature abstraction without anchoring. I had learned the rules before I had learned the game they belonged to. So I checked everything locally, step by step, never quite trusting the global invariant that made those steps safe.
Coming back to these problems now feels like closing a loop. I’m doing the same computations, but they compress differently. What once required vigilance now feels almost obvious. Not because the mathematics has become trivial, but because the framework has finally aligned with how I think.
This has been a recurring pattern in my learning. I often encounter ideas early, work with them awkwardly, and move on with a sense of unfinished business. Months later, after other structures have fallen into place, I return and realize that the difficulty was never the idea itself. It was my orientation to it.
Seeing this happen with something as concrete as modular exponentiation has been oddly reassuring. It’s a reminder that struggling with basic concepts doesn’t mean they are beyond reach. Sometimes it just means the right mental picture hasn’t arrived yet. And when it does, the mathematics doesn’t change. Only the way it sits in the mind does.
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