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Number Theory — The Queen of Mathematics

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...

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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.

Key sections

  • 01Number Theory
  • 02The Integers
  • 03The Primes are Infinite
  • 04Every Integer is Prime, Uniquely
  • 05How to Catch a Prime
  • 06The Prime Number Theorem
  • 07Modular Arithmetic
  • 08Fermat's Little Theorem
  • 09Diophantine Equations
  • 10Fermat's Last Theorem
  • 11Cryptography & Beyond
  • 12What We Do Not Know
  • 13Further Reading

Topics covered

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Slide outline
  1. 01Number Theory
  2. 02The Integers
  3. 03The Primes are Infinite
  4. 04Every Integer is Prime, Uniquely
  5. 05How to Catch a Prime
  6. 06The Prime Number Theorem
  7. 07Modular Arithmetic
  8. 08Fermat's Little Theorem
  9. 09Diophantine Equations
  10. 10Fermat's Last Theorem
  11. 11Cryptography & Beyond
  12. 12What We Do Not Know
  13. 13Further Reading
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Updated
2026-05-17
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Presentation Transcript

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Slide 01

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 02

§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 03

§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 04

§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 05

§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 06

§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 07

§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 08

§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 09

§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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