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Pure mathematics — no exam syllabus
Number Theory — The Queen of Mathematics
Gauss called it the "Queen of Mathematics." Number theory studies the properties of integers — the simplest, most fundamental objects in math. Yet it contains some of the deepest unsolved problems. This is the world of Ramanujan: patterns no one else saw, formulas that seemed to come from another dimension.
"Mathematics is the queen of sciences and number theory is the queen of mathematics."
— Carl Friedrich Gauss
Part 1
🔢 Prime Numbers — The Atoms of Arithmetic
The Building Blocks
Why Every Number Has a Unique Fingerprint
Every integer can be broken down into prime factors, like molecules breaking into atoms. 60 = 2² × 3 × 5. 84 = 2² × 3 × 7. No matter how you start factoring, you always get the same primes — the Fundamental Theorem of Arithmetic. This uniqueness is why primes are the foundation of all number theory. A prime number is one that can't be factored into smaller integers — it's "primeval," first-made.
Primes are the indivisible atoms of the number system. Every number has a unique prime factorization.
The Infinitude of Primes
Are there infinitely many primes? Euclid proved yes, around 300 BC, with breathtaking elegance:
Suppose primes are finite. List them all: p₁, p₂, ..., pₙ.
Consider N = (p₁ × p₂ × ... × pₙ) + 1.
N is not divisible by any pᵢ (each leaves remainder 1).
So either N is prime (and not in our list), or N has a prime factor not in our list.
Contradiction. Therefore primes are infinite.
This proof is considered one of the most beautiful in all of mathematics.
The Prime Distribution — A Deep Mystery
Although primes are infinite, they become rarer as numbers grow. The Prime Number Theorem says the density of primes near x is about 1/ln(x).
Prime Number Theorem
π(x) ~ x / ln(x)
The number of primes ≤ x is approximately x divided by the natural log of x.
| x | 10 | 100 | 1,000 | 10,000 | 1,000,000 | 10⁹ |
| π(x) | 4 | 25 | 168 | 1,229 | 78,498 | 50,847,534 |
The Great Unsolved Mystery
The Riemann Hypothesis — $1,000,000 Problem
The Prime Number Theorem tells us the average distribution of primes. But how far can actual prime counts deviate from this average? The answer lies in the zeros of the Riemann zeta function ζ(s). Riemann conjectured in 1859 that all non-trivial zeros have real part exactly ½. If true, it would reveal the deepest order in prime distribution. It remains unproven — one of the seven Clay Millennium Problems ($1M prize). The Riemann Hypothesis is arguably the most important unsolved problem in mathematics.
The pattern of primes is connected to the zeros of a complex function in a way we still don't fully understand.
Part 2
⏰ Modular Arithmetic — Clock Math
Math on a Clock
Gauss's Discovery at Age 20
Gauss published his masterpiece Disquisitiones Arithmeticae at age 24. In it, he introduced modular arithmetic: if it's 9 o'clock now, what time in 5 hours? 9 + 5 = 14 ≡ 2 (mod 12). The clock "wraps around." This is arithmetic modulo 12. Gauss called it "congruence" — two numbers are congruent mod n if they differ by a multiple of n. This simple idea is the foundation of modern cryptography, error-correcting codes, and digital security.
Modular arithmetic = arithmetic on a circle. Numbers wrap around when they reach n.
Congruence — The Language of Modular Arithmetic
We write a ≡ b (mod n) if n divides (a − b). Examples: 17 ≡ 5 (mod 12), 100 ≡ 1 (mod 9), 42 ≡ 0 (mod 7).
Fermat's Little Theorem
One of the most beautiful and useful results in number theory:
Fermat's Little Theorem
If p is prime and a is not divisible by p, then ap−1 ≡ 1 (mod p)
Example: 2¹⁰ = 1024 ≡ 1 (mod 11). Try it: 1024 ÷ 11 = 93 × 11 + 1 ✓
Consider arranging a beads in a circle of p positions. Rotating the circle gives p different arrangements (unless the pattern repeats). Since p is prime, the only patterns that repeat are the single-color ones (all a choices). So total arrangements ap minus the a monochromatic ones is divisible by p: ap − a = a(ap−1 − 1) ≡ 0 (mod p).
If a is not divisible by p, then ap−1 ≡ 1 (mod p).
Elegant combinatorial proof!
Euler's Theorem — Fermat's Generalization
Euler generalized Fermat's theorem to any modulus n using Euler's totient function φ(n), which counts numbers less than n that are coprime to n.
Part 3
🧩 Diophantine Equations — Solving for Integers
The Puzzle That Took 358 Years
Fermat's Last Theorem
In the 1630s, Fermat scribbled in a margin: "It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." He claimed xⁿ + yⁿ = zⁿ has no integer solutions for n > 2. The proof was lost — or perhaps he never had it. For 358 years, mathematicians tried and failed. Andrew Wiles finally proved it in 1994, using 20th-century mathematics Fermat couldn't have imagined: modular forms, elliptic curves, Galois representations. The margin was definitely too small.
The simplest-looking problems in number theory can require the deepest mathematics to solve.
Pythagorean Triples
The equation x² + y² = z² has infinitely many integer solutions — Pythagorean triples. The smallest: (3,4,5), (5,12,13), (8,15,17).
Generating All Triples
a = m² − n², b = 2mn, c = m² + n² (for any m > n)
Example: m=3, n=1 → (3²−1², 2×3×1, 3²+1²) = (8, 6, 10) = 2×(4,3,5)
Pell's Equation
The equation x² − Dy² = 1 (where D is not a perfect square) has infinitely many solutions, generated by the fundamental solution and its powers.
Pell's Equation: x² − 2y² = 1
Fundamental solution: (3,2) since 3² − 2×2² = 9 − 8 = 1.
All solutions come from powers of (3+2√2): (3+2√2)² = 17+12√2 → (17,12) → 17² − 2×12² = 289 − 288 = 1.
This equation appears in Archimedes' cattle problem and modern cryptography.
Part 4
🔐 Cryptography — Number Theory Protects the Internet
The Internet Runs on Primes
How Number Theory Secures Your Credit Card
Every time you buy something online, your browser and the server perform a cryptographic handshake. The security rests on one fact: multiplying two large primes is easy, but factoring the product is extremely hard. This asymmetry — easy one way, hard the other — is the basis of RSA encryption. Your browser uses modular exponentiation (Euler's theorem) to encrypt, and only the server's private key can decrypt. Without number theory, e-commerce as we know it couldn't exist.
The purest branch of mathematics turned out to have the most practical application imaginable.
RSA in a Nutshell
Key generation:
1. Pick two large primes p, q (hundreds of digits).
2. Compute n = p × q. φ(n) = (p−1)(q−1).
3. Choose e such that gcd(e, φ(n)) = 1 (typically 65537).
4. Compute d such that e × d ≡ 1 (mod φ(n)) — the modular inverse.
Public key: (n, e) — anyone can see this.
Private key: d — only the recipient knows this.
Encryption: c ≡ mᵉ (mod n)
Decryption: m ≡ cᵈ (mod n)
Why it works: By Euler's theorem, mφ(n) ≡ 1 (mod n). Since e×d = 1 + k×φ(n), we get cᵈ = me×d = m × mk×φ(n) ≡ m (mod n).
Why it's secure: To find d from (n,e), you need φ(n), which requires factoring n = p×q. Factoring large numbers is believed to be computationally infeasible.
Ramanujan's Legacy
Srinivasa Ramanujan (1887-1920) had no formal training but produced thousands of breathtaking results. He saw patterns in numbers that no one else could see. The Ramanujan tau function, Ramanujan's master theorem, Ramanujan-Nagell equation, and his work on continued fractions and modular forms continue to inspire modern number theory. His "taxicab number" — 1729 — is the smallest number expressible as a sum of two cubes in two different ways: 1729 = 1³ + 12³ = 9³ + 10³.
"An equation for me has no meaning unless it expresses a thought of God."
— Srinivasa Ramanujan
The Millennium Problems
Number theory still holds some of the deepest unsolved problems. Beyond the Riemann Hypothesis ($1M):
- Birch and Swinnerton-Dyer Conjecture — When does an elliptic curve have infinitely many rational points?
- ABC Conjecture — Proposed by Masser-Oesterlé, claimed by Mochizuki, relates to the radical of integers
- Twin Prime Conjecture — Are there infinitely many prime pairs (p, p+2)?
- Goldbach's Conjecture — Can every even number > 2 be written as sum of two primes?
🎯 Practice — Check Your Number Sense
These are not exam questions — they're puzzles that build number intuition.