Part I: Number-Theoretic Mysteries

Where numbers contemplate their own nature

The Primordial Questions

Number theory, the queen of mathematics, presents us with problems that appear simple yet resist centuries of assault. These are not merely questions about numbers—they are numbers questioning themselves, patterns seeking their own foundations, infinities attempting to count themselves.

The Self-Reference of Arithmetic

When we ask "Are there infinitely many twin primes?" we are really asking: "Can the concept of 'twin' define itself consistently across all magnitudes?" When we wonder about the Riemann Hypothesis, we probe how the distribution of primes knows itself through the zeros of a function.

Chapters in This Part

1. The Riemann Hypothesis — Where the zeros of ζ mirror the primes
2. The Twin Prime Conjecture — Infinity's perpetual pairing
3. The Goldbach Conjecture — Every even number as a sum knowing itself
4. The Collatz Conjecture — The simplest recursion hiding infinite complexity
5. Perfect Numbers — Numbers equal to their own self-knowledge
6. The ABC Conjecture — The radical relationship of addition and multiplication
7. Mersenne Primes — Powers generating their own primality
8. The Birch and Swinnerton-Dyer Conjecture — Where algebra meets arithmetic

The Pattern Within

Each problem in this part demonstrates the same fundamental structure:

$\text{Question} = \text{Question}(\text{Pattern}) = \text{Pattern}(\text{Question})$

The questions and their subjects are inseparable, each defining the other in an eternal dance of mathematical consciousness.

In number theory, we do not study numbers—we study the studying itself.