Fermat's Little Theorem

Fermat’s Little Theorem says that if $p$ is a prime number and $a$ is not divisible by $p$, then

$$a^{p-1}\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}p)\mathrm{.}$$

For a long time, this statement felt clever but unanchored to me. It looked like a fact about large powers, or about numbers mysteriously wrapping around. I understood the manipulation, but not the inevitability. The result felt impressive in the way a trick feels impressive.

What finally shifted my understanding was realizing that the theorem is not about powers at all. It is about permutation.

When you work modulo a prime, the nonzero residues form a finite multiplicative world. Every element has an inverse. In that world, multiplication by any nonzero number does not create anything new. It simply rearranges what is already there.

If you take the set $\{1,2,\dots,p-1\}$ and multiply every element by $a$, you do not get a new set. You get the same elements in a different order. The structure is preserved. Nothing collapses. Nothing escapes. Multiplication acts as a permutation of a fixed system.

Once I saw that, exponentiation stopped looking like growth along a line. Repeated multiplication is just repeated rearrangement inside a closed space. Cycles are unavoidable, and the size of the system dictates when you return to where you started. The theorem stops feeling clever and starts feeling necessary.

Around the same time, I noticed something similar happening in my drawing practice.

In drawing, an early mistake in proportion or angle creates a broken structure. The natural response is to erase, abandon the piece, or start over. I used to do that often. Now I try to do the opposite. I finish the drawing anyway.

Once a mistake is made, nothing new can be added to erase it. The structure is fixed. All that remains is rearrangement. Line weight, shading, emphasis, and contrast begin to act like permutations. They do not remove the original error. They redistribute attention, balance, and form within the constraints that already exist.

The drawing becomes a closed system. You are no longer creating freely. You are rearranging.

This is where the connection became clear to me. In both mathematics and drawing, I was initially trying to escape constraints. I wanted the powers to behave nicely. I wanted the sketch to stay anatomically correct. When that failed, I felt stuck.

The breakthrough came from accepting the system as it was and working entirely inside it.

In Fermat’s Little Theorem, once you accept the finite multiplicative structure modulo a prime, the result follows automatically. In drawing, once you accept the broken proportions, the work becomes about problem-solving rather than correction. Progress can come not from adding something new, but from redistributing what is already there.

That is why I am comfortable sharing failed sketches alongside unfinished mathematical understanding. They are records of the same practice. Completion without denial. Learning without erasure. Respect for structure rather than frustration with it.

Neither the theorem nor the drawing is interesting because it looks polished. They matter because they show how much work structure does for you once you stop fighting it.

Also, here's my first serious hand sketch.