# Number Theory — The Queen of Mathematics

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Category: Mathematics
Slides: 13
Updated: 2026-05-17T20:55:52.301Z
Tags: catalog, math, number, theory

## Summary

2 3 “Mathematics is the queen of the sciences, and number theory is the queen of mathematics.” — C. F. Gauss ❦ ❦ Chapter I The Integers §1. The playing field The Integers The integers ℤ = { … , −3, −2, −1, 0, 1, 2, 3, … } form the bedrock of arithmetic — discrete, unbounded, equipped with addition and multiplication. Key sections include: Number Theory; The Integers; The Primes are Infinite; Every Integer is Prime, Uniquely; How to Catch a Prime; The Prime Number Theorem; Modular Arithmetic; Fermat's Little Theorem; Diophantine Equations; Fermat's Last Theorem.

## Slide Outline

1. Number Theory
2. The Integers
3. The Primes are Infinite
4. Every Integer is Prime, Uniquely
5. How to Catch a Prime
6. The Prime Number Theorem
7. Modular Arithmetic
8. Fermat's Little Theorem
9. Diophantine Equations
10. Fermat's Last Theorem
11. Cryptography & Beyond
12. What We Do Not Know
13. Further Reading

## Slide Transcript

### Slide 1: Number
Theory

- A Treatise on
- The Queen of Mathematics
- “Mathematics is the queen of the sciences, and number theory is the queen of mathematics.”
- — C. F. Gauss

### Slide 2: §1. The playing field

- Chapter IThe Integers
- The Integers
- The integers ℤ = { … , −3, −2, −1, 0, 1, 2, 3, … } form the bedrock of arithmetic — discrete, unbounded, equipped with addition and multiplication.
- Positive the natural numbers, ℕ
- Negative their additive inverses
- Zero the still center, an Indian gift to algebra
- Every non-empty subset of ℕ has a least element.

### Slide 3: §2. The atoms of arithmetic

- Chapter IIPrimes
- The Primes are Infinite
- A prime p > 1 has no divisors save 1 and itself. The first few:
- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …
- There exist infinitely many primes.
- Suppose, for contradiction, that the primes are finite: p₁, p₂, …, pn. Form N = p₁·p₂·⋯·pn + 1. Then N is divisible by some prime q. But q cannot equal any pi, for that would imply q ∣ 1. So q is a new prime — contradicting our list.
- A proof of crystalline economy — twenty-three centuries old, and still untouched.

### Slide 4: §3. The fundamental theorem

- Chapter IIIUnique Factorisation
- Every Integer is Prime, Uniquely
- Every integer n > 1 may be written as a product of primes n = p₁a₁ · p₂a₂ · ⋯ · pkak, and this factorisation is unique up to order.
- Examples — the unique decompositions:
- 12 = 2² · 3
- 360 = 2³ · 3² · 5
- 1001 = 7 · 11 · 13
- 2026 = 2 · 1013
- Fig. 1The factor tree of 360.

### Slide 5: §4. The sieve and its descendants

- Chapter IVFinding Primes
- How to Catch a Prime
- Eratosthenes, c. 240 B.C. — the librarian of Alexandria. Strike out multiples of 2, then 3, then 5… what remains is prime.
- Trial Division — test divisors up to √n. Honest, slow.
- Miller–Rabin, 1976 — a probabilistic test exploiting Fermat's little theorem. Practically certain in microseconds.
- AKS, 2002 — deterministic polynomial-time primality. A theoretical jewel.
- Fig. 2The sieve to 30, with an Ulam spiral.

### Slide 6: §5. The density of primes

- Chapter VDistribution
- The Prime Number Theorem
- Let π(n) denote the number of primes ≤ n. The primes thin out — but with breathtaking regularity.
- As n → ∞,
- π(n) ~ n / ln(n)
- that is, π(n) · ln(n) / n → 1.
- Fig. 3The prime counting function π(n) against its smooth approximation.

### Slide 7: §6. Clock arithmetic

- Chapter VICongruences
- Modular Arithmetic
- Two integers are congruent modulo n if their difference is divisible by n:
- a ≡ b (mod n) ⟺ n ∣ (a − b)
- So 17 ≡ 5 (mod 12) — both have remainder 5 when divided by 12. Time, days of the week, hours on a clock: all modular.
- Gauss formalised the notation in his 1801 masterwork Disquisitiones Arithmeticae, written at age 24 — and the subject was reborn.
- Fig. 4ℤ/12ℤ: the clock.

### Slide 8: §7. A jewel of congruence

- Chapter VIIFermat
- Fermat's Little Theorem
- Let p be prime. For every integer a,
- ap ≡ a (mod p)
- Equivalently, if gcd(a,p)=1, then ap−1 ≡ 1 (mod p).
- A small theorem with vast consequences. It underlies:
- The Miller–Rabin probabilistic primality test.
- The correctness of the RSA cryptosystem.
- Euler's generalisation: aφ(n) ≡ 1 (mod n).
- Fermat scribbled it in a letter; the first published proof came from Euler, a century later.

### Slide 9: §8. Equations in integers

- Chapter VIIIDiophantus
- Diophantine Equations
- A Diophantine equation demands integer solutions. The simplest non-trivial example:
- x² + y² = z²
- Pythagorean triples — (3,4,5), (5,12,13), (8,15,17), (7,24,25), … infinitely many, all generated by
- x = m²−n², y = 2mn, z = m²+n²
- for coprime m > n > 0 of opposite parity.
- Fig. 5The (3,4,5) Pythagorean triple.

### Slide 10: §9. Three and a half centuries

- Chapter IXFermat's Last Theorem
- Fermat's Last Theorem
- For any integer n > 2, the equation
- xn + yn = zn
- has no solutions in positive integers x, y, z.
- Fermat wrote in the margin of his Diophantus:
- “Cuius rei demonstrationem mirabilem sane detexi, hanc marginis exiguitas non caperet.” — I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.
- It withstood Euler, Sophie Germain, Kummer, generations of attack. In 1995, Andrew Wiles — after seven years of secret labour — proved the modularity of semistable elliptic curves, from which Fermat's Last Theorem cascaded as a corollary.
- ❦ ❦ ❦

### Slide 11: §10. Number theory now

- Chapter XThe Modern Cathedral
- Cryptography & Beyond
- RSA & Public Key
- 1977 — Rivest, Shamir, Adleman. The hardness of factoring large semi-primes secures the world's banking, commerce, secrets. Number theory keeps your messages.
- Elliptic Curves
- Cubic curves y² = x³ + ax + b over finite fields. The arithmetic of their rational points fueled Wiles's proof and now powers ECC, used in TLS and blockchains.
- The Langlands Program
- Robert Langlands, 1967 — a vast web of conjectures linking number theory, representation theory, and harmonic analysis. A “grand unified theory” of mathematics, still being charted.
- Galois representations ↔ automorphic forms.
- The queen now wears digital robes.

### Slide 12: §11. Mysteries that abide

- Chapter XIOpen Problems
- What We Do Not Know
- All non-trivial zeros of ζ(s) = Σ 1/ns have real part ½. — A million-dollar Clay Prize. Equivalent to the deepest statements about the distribution of primes.
- There are infinitely many primes p such that p + 2 is also prime. Zhang (2013): infinitely many prime gaps less than 70 million. Polymath then drove the bound below 250.
- Every even integer n > 2 is the sum of two primes. Verified for n < 4 × 1018; proven for none.
- The simplest questions hide the deepest secrets. Such has always been her way.

### Slide 13: Further Reading

- ColophonReferences
- ❦ ❦ ❦
- Books
- G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers (1938).
- C. F. Gauss, Disquisitiones Arithmeticae (1801).
- K. Ireland & M. Rosen, A Classical Introduction to Modern Number Theory.
- S. Singh, Fermat's Enigma (1997).
- J. Derbyshire, Prime Obsession.
- Euclid, Elements, Books VII–IX.
- Video Lectures
- Fermat's Last Theorem & Andrew Wiles — the documentary and lectures.
- Prime Numbers, Explained — visual primers and proofs.
- A Final Word
- “God may not play dice with the universe, but something strange is going on with the prime numbers.” — Paul Erdős
- — Finis —


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