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
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, …
A proof of crystalline economy — twenty-three centuries old, and still untouched.
Examples — the unique decompositions:
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.
Let π(n) denote the number of primes ≤ n. The primes thin out — but with breathtaking regularity.
Two integers are congruent modulo n if their difference is divisible by n:
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.
A small theorem with vast consequences. It underlies:
Fermat scribbled it in a letter; the first published proof came from Euler, a century later.
A Diophantine equation demands integer solutions. The simplest non-trivial example:
Pythagorean triples — (3,4,5), (5,12,13), (8,15,17), (7,24,25), … infinitely many, all generated by
for coprime m > n > 0 of opposite parity.
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.
1977 — Rivest, Shamir, Adleman. The hardness of factoring large semi-primes secures the world's banking, commerce, secrets. Number theory keeps your messages.
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.
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.
The queen now wears digital robes.
The simplest questions hide the deepest secrets. Such has always been her way.
“God may not play dice with the universe, but something strange is going on with the prime numbers.” — Paul Erdős
— Finis —