Number theory.

OpeningThe queen of mathematics.

"Mathematics is the queen of the sciences," Gauss wrote, "and number theory is the queen of mathematics." The compliment was not idle. Gauss devoted his greatest energies to it.

The numbers seem the simplest objects in mathematics — 1, 2, 3, 4 — and the questions about them are the deepest. Whether infinitely many twin primes exist is unsolved. Whether the digits of π are uniformly distributed is unsolved. Whether the most famous unsolved problem in mathematics — the Riemann hypothesis — is true, is unsolved.

This deck traces the queen's portrait. From Euclid's proof that there are infinitely many primes to Wiles's proof of Fermat's last theorem; from Ramanujan's notebooks to Viazovska's solution to sphere packing; the longest-running act in mathematics.

Chapter IEuclid on primes.

Books VII–IX of Euclid's Elements (c. 300 BCE) contain the first systematic number theory. Definitions of prime, composite, and even and odd; the Euclidean algorithm for greatest common divisors; the proof that there are infinitely many primes.

The infinitude proof is one of the most beautiful arguments in mathematics. Suppose only finitely many primes p₁, p₂, …, p_n exist. Form the number N = p₁p₂…p_n + 1. Then N is either prime itself or has a prime factor — but it cannot be divisible by any of p₁, …, p_n, since dividing leaves remainder 1. So a prime is missing from the list. Contradiction.

The proof was reproduced by G. H. Hardy in A Mathematician's Apology (1940) as exhibit A in the case for mathematical beauty. Twenty-three centuries on, no improvement on Euclid's argument has been found.

Chapter IIDiophantine equations.

Diophantus of Alexandria (c. 200–284 CE) wrote the Arithmetica, thirteen books on integer and rational solutions to polynomial equations. Six books survive in Greek; four more in 1968 turned up in an Arabic translation in Mashhad, Iran.

Diophantus introduced "syncopated" abbreviations for the unknown and its powers — between rhetorical word-arithmetic and modern symbolic notation. His problems do not seek all solutions to a given equation but a single illustrative one, often by ingenious substitutions that have echoed through the field.

The term Diophantine equation now means any polynomial equation for which we seek integer or rational solutions. x² + y² = z² (Pythagorean triples) is the basic case; x^n + y^n = z^n for n > 2 (Fermat) is the famous case. Hilbert's tenth problem, posed in 1900, asked for an algorithm to decide solvability of Diophantine equations; Yuri Matiyasevich proved in 1970 no such algorithm exists.

Chapter IIIFermat's little theorem.

If p is prime and a is not divisible by p:

a^(p−1) ≡ 1 (mod p).

Pierre de Fermat stated the theorem in a 1640 letter to Bernard Frénicle de Bessy. He claimed a proof but, characteristically, did not write it down. The first published proof was given by Euler in 1736.

The result is the foundation of modern primality testing and of the RSA cryptosystem. The Miller–Rabin probabilistic primality test asks whether a^(n−1) ≡ 1 (mod n) for many random a; if not, n is composite. (The converse is almost — but not quite — true, as Carmichael numbers attest.) The same identity drives RSA encryption: messages are recovered through exponentiation modulo a prime product.

Chapter IV1637 conjecture, 1995 proof.

Around 1637, Fermat wrote in the margin of his copy of Diophantus's Arithmetica:

"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 a truly marvellous demonstration of this, which this margin is too narrow to contain."

For 358 years no proof — Fermat's or anyone else's — was found. Many great mathematicians attempted it. Sophie Germain made the first major progress (early 1800s); Ernst Kummer developed the theory of ideals working on it (1840s); generations of partial results piled up, but the general statement resisted.

The connection to elliptic curves emerged in 1985: Gerhard Frey observed that a counterexample to Fermat would produce an elliptic curve too strange to satisfy the Taniyama–Shimura modularity conjecture. Ken Ribet made the connection rigorous in 1986. The path to Fermat now ran through modularity.

Chapter VAndrew Wiles.

Andrew Wiles (b. 1953) had been obsessed with Fermat since age 10. After Ribet's 1986 connection, he saw a path. He worked in secret in his Princeton attic for seven years.

In June 1993 he announced his proof at the Isaac Newton Institute in Cambridge — three lectures, the result revealed at the end of the third. Within months a gap was found. For another year Wiles, with his former student Richard Taylor, struggled to repair it.

The repair came on 19 September 1994. Wiles described it as "the most important moment of my working life." The complete proof was published in two papers in Annals of Mathematics in 1995: Wiles's "Modular elliptic curves and Fermat's last theorem" and the joint Taylor–Wiles paper closing the gap.

He was knighted in 2000, awarded the Abel Prize in 2016. Fermat's marginal note remains the most successful publicity stunt in mathematical history.

Chapter VIGoldbach's conjecture.

In a June 7, 1742 letter to Euler, Christian Goldbach conjectured (in the modern reformulation): every even integer greater than 2 is the sum of two primes.

Verified by computer for every even integer up to 4 × 10¹⁸ (Oliveira e Silva, 2014). Believed true. Unproved.

The weak Goldbach conjecture — every odd integer greater than 5 is the sum of three primes — was proved by Harald Helfgott in 2013, building on earlier results of Vinogradov (1937, asymptotically) and computer-verified by Helfgott et al. for the small cases.

The strong (binary) Goldbach is one of the oldest unsolved problems in number theory. It would follow from suitable strengthenings of the prime number theorem in arithmetic progressions, but those strengthenings are themselves open. The conjecture is the type of result that "should" be true and may resist proof for a long time yet.

Chapter VIIThe twin prime conjecture.

Are there infinitely many primes p such that p + 2 is also prime? The conjecture has been around since Euclid in spirit; Alphonse de Polignac stated it in 1849.

The largest known twin prime pair has thousands of digits and is found via large-scale distributed computation. The conjecture itself remained completely out of reach until 2013, when Yitang Zhang — a 58-year-old lecturer at the University of New Hampshire who had subsisted for years on adjunct positions — proved that there are infinitely many prime pairs with gap less than 70 million.

It was the first finite bound ever proved on prime gaps. The Polymath8 project, organised online via Terence Tao's blog, drove the bound down to 4,680 within months. James Maynard's independent technique reduced it further to 600, and the project is now stuck at 246 (assuming the Elliott–Halberstam conjecture, the bound drops to 12; the conjecture itself, gap 2, remains open).

Chapter VIIIMersenne primes.

A Mersenne number has the form M_p = 2^p − 1 for prime p. When M_p is itself prime, it is a Mersenne prime. Marin Mersenne (1588–1648) listed candidates for prime exponents up to 257; he made errors, but the name stuck.

The Lucas–Lehmer test gives a simple, fast deterministic primality test specific to Mersenne candidates. This makes them the easiest source of huge primes — and hence the prime-finding world's record holders.

As of 2024, 52 Mersenne primes are known. The largest is M_{136,279,841}, discovered in October 2024 by Luke Durant via the Great Internet Mersenne Prime Search (GIMPS) — a 41-million-digit number. Whether infinitely many Mersenne primes exist is unproven.

Chapter IXPerfect numbers.

A perfect number equals the sum of its proper divisors. The first four: 6 = 1 + 2 + 3; 28 = 1 + 2 + 4 + 7 + 14; 496; 8128.

Euclid proved (Book IX, Proposition 36): if 2^p − 1 is prime, then 2^(p−1)(2^p − 1) is perfect. Euler (1849, posthumous) proved the converse for even perfect numbers — every even perfect number has Euclid's form.

So the even perfect numbers are in exact correspondence with the Mersenne primes. There are currently 52 even perfect numbers known; whether infinitely many exist is the same question as whether infinitely many Mersenne primes exist.

Whether any odd perfect number exists is unknown. None has been found below 10¹⁵⁰⁰. The conjecture is that none exists. The result, if proved, would rank among the great theorems of elementary number theory.

Chapter XThe Riemann hypothesis.

The most famous open problem in mathematics. Bernhard Riemann's 1859 paper "On the Number of Primes Less Than a Given Magnitude" — his only paper on number theory, eight pages — introduced the zeta function as a function of a complex variable and conjectured: all non-trivial zeros of ζ(s) have real part 1/2.

The conjecture controls the distribution of prime numbers with extreme precision. The prime number theorem (the count of primes up to x is approximately x / log x) is equivalent to: no zero of ζ on the line Re(s) = 1. The Riemann hypothesis would give the strongest possible error bound.

Numerical evidence: every one of the first 10¹³ non-trivial zeros lies on the critical line. The hypothesis is one of the seven Clay Millennium Prize Problems (US$1 million for a proof). It has been a target for 165 years.

Chapter XIThe zeta function.

Euler studied the function ζ(s) = 1 + 1/2^s + 1/3^s + 1/4^s + … for real s > 1. He proved ζ(2) = π²/6 in 1734, settling the Basel problem that had stumped the Bernoullis for decades.

Euler also discovered the Euler product formula:

ζ(s) = ∏_p (1 − p^(−s))^(−1), the product over all primes.

The identity is profound. The infinite sum and infinite product are equal, expressing the unique-factorisation theorem analytically. It connects the additive structure of the integers (the sum on the left) to the multiplicative structure of the primes (the product on the right). All of analytic number theory lives in the consequences of this single identity.

Riemann's contribution was to extend ζ to a meromorphic function of a complex variable and study its zeros. The continuation passes through famous values: ζ(−1) = −1/12; ζ(0) = −1/2.

Chapter XIIThe prime number theorem.

The number π(x) of primes up to x satisfies π(x) ~ x / log x as x → ∞.

The conjecture was made independently by Gauss (age 15, in his diary) and Legendre (1798). It was proved more than a century later, simultaneously and independently, by Jacques Hadamard and Charles-Jean de la Vallée Poussin in 1896. Both proofs used Riemann's machinery — complex analysis applied to ζ(s) on the line Re(s) = 1.

An elementary proof — one not using complex analysis — was found by Atle Selberg and Paul Erdős in 1948, settling a bet (Hardy had insisted no elementary proof was possible). The work led to a memorable priority quarrel; both eventually received Fields-equivalent honours, Selberg the Fields Medal in 1950 and Erdős the Wolf Prize in 1983.

The error in the approximation π(x) ≈ x / log x is intimately connected to the Riemann hypothesis: better hypotheses on the zeros of ζ give smaller error bars.

Chapter XIIIHardy and Ramanujan.

In January 1913 a 25-year-old clerk in the Madras Port Trust office sent a letter, ten pages of unfamiliar formulas, to G. H. Hardy at Cambridge. Hardy and his collaborator J. E. Littlewood spent an evening on it. Hardy concluded: "They must be true, because if they were not true, no one would have had the imagination to invent them."

Srinivasa Ramanujan (1887–1920) arrived in Cambridge in April 1914. The collaboration with Hardy, lasting until Ramanujan's return to India in 1919 and his death in 1920, produced major work on partition theory, modular forms, and asymptotic analysis. The Hardy–Ramanujan formula for the number of partitions of n is among the most beautiful asymptotic results in mathematics.

Their joint creation of the circle method in analytic number theory remains the standard tool for many additive problems, including Vinogradov's three-prime theorem and modern attacks on the Goldbach conjecture.

Chapter XIVRamanujan's notebooks.

Ramanujan filled three large notebooks with formulas — without proofs — between 1903 and 1914. After his death, a "lost notebook" of 138 pages was discovered in 1976 by George Andrews in the Wren Library at Trinity College, Cambridge. It contained material on mock theta functions and other topics from the last year of his life.

The notebooks have been mined for a century by mathematicians who fill in the proofs and discover the meaning of the identities. Bruce Berndt's five-volume Ramanujan's Notebooks (1985–1998) painstakingly worked through the unproved earlier notebooks. The lost notebook is still being unpacked.

The mock theta functions Ramanujan introduced in his last letter to Hardy (January 1920) found their place in modern mathematics in 2002, when Sander Zwegers's thesis showed they were the holomorphic parts of harmonic Maass forms. The connection has reinvigorated modular form theory.

Chapter XVModular forms.

A modular form is a holomorphic function on the upper half-plane that transforms in a specific way under the modular group SL(2, Z) — the integer 2×2 matrices with determinant 1.

The simplest examples are Eisenstein series and the discriminant function Δ(z). The space of modular forms of given weight is finite-dimensional; this surprising fact yields powerful identities — many number-theoretic results are obtained by writing a function as a modular form and using the dimension formula.

The Langlands philosophy positions modular forms as the analytic shadows of Galois representations. Wiles's proof of Fermat's last theorem proceeds by associating a modular form to every elliptic curve, using the Taniyama–Shimura modularity conjecture as a bridge. Maryam Mirzakhani, Akshay Venkatesh, and many recent Fields medallists have worked at the modular-form / arithmetic intersection.

Chapter XVIElliptic curves.

An elliptic curve is a smooth cubic curve y² = x³ + ax + b (with non-zero discriminant). Despite the name, they are not ellipses; the name comes from the elliptic integrals that arose computing arc lengths of ellipses.

The points on an elliptic curve form an abelian group under a geometrically defined addition (the chord-and-tangent law). This is the surprising fact that drives the subject — a curve carries a group, and the group's structure encodes deep arithmetic information.

The Mordell theorem (1922): the rational points on an elliptic curve over Q form a finitely-generated abelian group. The rank — the number of independent infinite-order generators — is generally hard to compute and is the subject of the Birch and Swinnerton-Dyer conjecture (next leaf).

Elliptic curves are now central to cryptography: elliptic-curve cryptography (Koblitz and Miller, independently, 1985) gives equivalent security to RSA at much smaller key sizes and is the standard for modern protocols like TLS and Signal.

Chapter XVIIThe modularity theorem.

Every elliptic curve over Q is modular — it corresponds to a modular form of weight 2.

The conjecture was made in the 1950s, in slightly different forms, by Yutaka Taniyama and Goro Shimura. Taniyama died by suicide in 1958 at age 31. Shimura developed the conjecture into a precise form that became central to modern number theory.

Andrew Wiles's 1995 proof established modularity for the semistable case — sufficient to imply Fermat's last theorem. The full theorem was proved in 2001 by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor.

The modularity theorem is a special case of the larger Langlands philosophy. It is the most spectacular instance to date of the dictionary between Galois representations and automorphic forms.

Chapter XVIIIBirch and Swinnerton-Dyer.

One of the seven Clay Millennium Prize Problems. The conjecture connects two faces of an elliptic curve: the rank of its group of rational points (an algebraic invariant) and the order of vanishing of its L-function at s = 1 (an analytic invariant).

Bryan Birch and Peter Swinnerton-Dyer formulated the conjecture in the 1960s based on extensive computer experiments at Cambridge — one of the earliest examples of computational mathematics shaping a major conjecture.

Partial results: Coates–Wiles (1977), Gross–Zagier (1986), Kolyvagin (1988) established the conjecture in rank 0 and 1 cases. Higher ranks remain open. The conjecture would, if proved, give a complete algorithmic procedure for finding the rank of any elliptic curve.

Chapter XIXAlgebraic number theory.

The study of algebraic numbers — roots of polynomials with integer coefficients — and the rings of integers of number fields. Gauss's 1801 Disquisitiones Arithmeticae initiated the modern subject; the Gaussian integers Z[i] were one early example.

Ernst Kummer, attempting Fermat's last theorem in the 1840s, discovered that unique factorisation can fail in rings of algebraic integers. He invented "ideal numbers" to restore it — formal divisors that don't correspond to elements of the ring. Richard Dedekind recast the theory in terms of ideals (1871), giving the modern framework.

David Hilbert's 1897 Zahlbericht ("Number Report") was the canonical reference for a generation. Emil Artin, Helmut Hasse, and Hermann Weyl developed class field theory in the 1920s–30s. The subject is now the technical foundation of modern algebraic number theory and a major theme of the Langlands program.

Chapter XXClass field theory.

Class field theory describes the abelian extensions of a number field — extensions whose Galois group is abelian — in terms of internal arithmetic data of the field itself.

The classical case (Hilbert 1898): every number field has a unique maximal unramified abelian extension, the Hilbert class field. Its Galois group is canonically the field's ideal class group, which measures the failure of unique factorisation in the ring of integers.

The modern formulation, via Artin reciprocity (Emil Artin, 1927), generalises classical results from quadratic and cubic reciprocity (which Gauss called the "queen of theorems," reproving it eight times). Class field theory is to number theory what Galois theory is to field theory: the symmetry framework that organises the deepest results.

Chapter XXIL-functions.

A class of meromorphic functions of a complex variable that generalises the Riemann zeta function. The simplest examples are Dirichlet L-functions, introduced by Lejeune Dirichlet in 1837 to prove that every arithmetic progression with first term coprime to common difference contains infinitely many primes.

L-functions attach to many number-theoretic objects: number fields, elliptic curves, modular forms, Galois representations, automorphic representations. The unifying conjecture, originating with Langlands, predicts that all "naturally occurring" L-functions are essentially automorphic.

L-functions are the analytic engine of modern number theory. The Riemann hypothesis generalises naturally: the generalised Riemann hypothesis (GRH) asserts that every Dirichlet L-function has its non-trivial zeros on the critical line. Many conditional results — including the strongest current bounds on prime distribution in short intervals and progressions — assume GRH.

Chapter XXIICryptography.

Number theory was, in Hardy's 1940 Apology, the most useless of mathematical subjects: "real mathematics has no effects on war." The judgement aged poorly.

RSA (Rivest, Shamir, Adleman, 1977) is a public-key cryptosystem whose security rests on the difficulty of factoring large semiprimes. Encryption uses modular exponentiation in (Z/nZ)*; the Euler totient φ(n) drives decryption. Knowing the factorisation of n gives φ(n) directly; not knowing it costs years of computation.

Elliptic-curve cryptography (Koblitz, Miller, 1985) achieves equivalent security with much smaller key sizes by working in the group of an elliptic curve over a finite field. Modern protocols — TLS 1.3, Signal, Bitcoin signatures — use elliptic curves rather than RSA. Number theory secures the global financial system.

Chapter XXIIIThe quantum threat.

In 1994 Peter Shor at Bell Labs published a quantum algorithm that factors integers in polynomial time. The same algorithm, suitably modified, computes discrete logarithms in elliptic-curve groups in polynomial time.

If a sufficiently large fault-tolerant quantum computer is ever built, RSA, Diffie–Hellman, and elliptic-curve cryptography all break. The current best estimate is that several thousand logical qubits would be required; the largest current quantum computers have a few hundred noisy physical qubits, and progress is rapid but incremental.

The cryptographic response is post-quantum cryptography: schemes whose security rests on problems believed hard even for quantum computers — lattice problems, code-based problems, hash-based signatures. NIST standardised the first post-quantum algorithms (CRYSTALS-Kyber, CRYSTALS-Dilithium, SPHINCS+, FALCON) in 2024. The transition is one of the largest cryptographic migrations in history.

Chapter XXIVThe Langlands program.

The grandest organising vision in modern number theory. Robert Langlands's 1967 letter to André Weil proposed a vast web of correspondences between Galois representations of number fields (the arithmetic side) and automorphic representations of reductive algebraic groups (the analytic side).

The simplest case — the modularity of elliptic curves over Q — is the Wiles–Taylor–Breuil–Conrad–Diamond theorem behind Fermat. The function-field analogue (the Langlands conjecture for GL_n over function fields) was proved by Laurent Lafforgue (Fields Medal 2002) and extended by Vincent Lafforgue (Breakthrough Prize 2019).

The geometric Langlands program — Drinfeld, Beilinson, Gaitsgory — connects the program to representation theory and gauge theory via mirror symmetry. Dennis Gaitsgory and collaborators announced a proof of the geometric Langlands conjecture in May 2024 — the largest single result in the program's history.

Chapter XXVViazovska, 2016.

The densest packing of equal spheres. The 3D answer (orange-stand arrangement) was conjectured by Kepler in 1611 and proved by Thomas Hales in 1998 (with computer verification completed in 2014).

In 2016, Maryna Viazovska, a 31-year-old Ukrainian mathematician, published an astonishing 22-page paper proving that the densest sphere packing in 8 dimensions is the E₈ lattice. A week later, with collaborators, she extended the result to 24 dimensions (the Leech lattice).

The proof uses modular forms in an unexpected way: she constructs an explicit "magic function" via theta-series machinery whose Fourier transform encodes both an upper bound on packing density and the lower bound achieved by E₈. The bounds match exactly.

The proof is universally regarded as one of the most beautiful results of the 21st century. Viazovska won the Fields Medal in 2022, the second woman to do so (after Maryam Mirzakhani in 2014).

Chapter XXVITwenty essentials.

• c.300 BCEElements, Books VII–IXEuclid
• c.250ArithmeticaDiophantus
• 1801Disquisitiones ArithmeticaeGauss
• 1859On the Number of Primes…Riemann
• 1897ZahlberichtHilbert
• 1940A Mathematician's ApologyHardy
• 1957An Introduction to the Theory of NumbersHardy & Wright
• 1976A Course in ArithmeticSerre
• 1976Number TheoryBorevich & Shafarevich
• 1990The Man Who Knew InfinityKanigel
• 1992Algebraic Number TheoryNeukirch
• 1995Modular Elliptic Curves and Fermat's LastWiles
• 1997Fermat's EnigmaSingh
• 2000The Riemann Hypothesis (Clay)Bombieri
• 2003Prime ObsessionDerbyshire
• 2003The Music of the Primesdu Sautoy
• 2009The Arithmetic of Elliptic CurvesSilverman
• 2013Bounded Gaps Between PrimesZhang
• 2017The Sphere Packing Problem in Dimension 8Viazovska
• 2020An Illustrated Theory of NumbersWeissman

Chapter XXVIIWatch & read.

↑ BBC Horizon · Fermat's Last Theorem · Andrew Wiles documentary

More on YouTube

Watch · The Riemann Hypothesis, explained
Watch · Numberphile · Gaps between primes

Chapter XXVIIIWhere to begin.

Start with Hardy and Wright's An Introduction to the Theory of Numbers (1938; many revisions, still in print). Slightly old-fashioned in style but unmatched in elegance. Niven, Zuckerman, and Montgomery's An Introduction to the Theory of Numbers is the modern standard textbook.

For history and atmosphere: Robert Kanigel's The Man Who Knew Infinity on Ramanujan; Simon Singh's Fermat's Enigma on Wiles; Marcus du Sautoy's The Music of the Primes on the Riemann hypothesis.

For the modern pure subject: Serre's A Course in Arithmetic is short, dense, and contains more per page than almost any other math book. Silverman's Arithmetic of Elliptic Curves is the standard graduate introduction to its subject.

The reward of going deep: number theory is the sub-discipline where mathematics most often surprises with its beauty. The objects are simple. The depth is bottomless.