Number Theory as a Sherlock-Style Investigation

 One pattern that has become clear in my number theory work is that many problems behave like small detective cases. Not in a poetic sense, but in a very literal mathematical way: you begin with constraints, eliminate impossible candidates, and narrow down the set of integers that can satisfy all conditions simultaneously.

A simple example is solving something like

$$x\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}7)\mathrm{<br>}$$

The complete solution set is

$$\{1 + 7k : k \in \mathbb{Z}\}.$$

At first this looked like a formula to memorize, but the more I worked with congruences, the more this felt like identifying a “suspect class.” Out of all integers, only those that remain 1 after reduction mod 7 qualify. The modulus eliminates most of the search space immediately, which is exactly what a good constraint is supposed to do.

When the problems get layered, the detective analogy becomes even clearer. For instance, suppose I want all integers $x$ satisfying both

$$x\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}5)\text{and}x\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)\mathrm{<br>}$$

Each condition describes a different arithmetic progression:

• $x = 2 + 5k$
  

• $x = 3 + 4m$
  

Finding the intersection is basically matching two patterns until they coincide. In practice, this means substituting one into the other and solving the resulting linear congruence. Mathematically it is routine, but the process genuinely feels like checking alibis. Each modulus rules out a large portion of numbers, and only those that fit every condition survive.

This viewpoint also helped me understand why methods like the Chinese Remainder Theorem work. CRT essentially guarantees that if the constraints are consistent and the moduli are coprime, then exactly one residue class satisfies all of them. That is a very strong structural statement. It tells you that these “clues” are not arbitrary; they mesh perfectly because the arithmetic structure forces them to.

The more problems I solve, the more I rely on this way of thinking. Instead of asking “What is the answer?”, I ask “Which integers are even allowed to be the answer?” Once the constraints carve out the space, the solution usually becomes straightforward.

This shift has made number theory feel less like computation and more like systematic elimination. And now that I have noticed it, I am starting to approach new problems by looking for the quickest way to restrict the search space. It is a surprisingly effective strategy, especially for congruences and Diophantine equations.

Also, I know I diverged a bit from the order of chapters in Silverman but I'm still going slowly and carefully.