# Chronology of Number Theory

## Ancient Mathematics

| Period | Development |
|---|---|
| c. 1800 BCE | Babylonian arithmetic tables and quadratic problems |
| c. 1650 BCE | Egyptian arithmetic in the Rhind Papyrus |
| c. 500 BCE | Early Greek studies of ratios and integers |
| c. 300 BCE | entity["people","Euclid","ancient Greek mathematician"] writes Elements, including Euclidean algorithm and infinitude of primes |
| c. 250 BCE | Study of perfect numbers and geometric arithmetic |

## Classical and Late Ancient Era

| Period | Development |
|---|---|
| c. 250 CE | entity["people","Diophantus","ancient Greek mathematician"] studies rational and integer equations |
| c. 400 CE | Arithmetic commentaries preserve Greek number theory |
| c. 600-1200 | Indian and Islamic mathematicians advance algebra and arithmetic methods |

## Early Modern Number Theory

| Period | Development |
|---|---|
| 1600s | entity["people","Pierre de Fermat","French mathematician"] develops descent, congruences, and Fermat problems |
| 1640 | Fermat states Fermat’s Last Theorem |
| 1657 | Fermat studies sums of two squares |
| Late 1600s | Development of symbolic algebra and infinite series |

## Eighteenth Century

| Period | Development |
|---|---|
| 1700s | entity["people","Leonhard Euler","Swiss mathematician"] introduces analytic methods into arithmetic |
| 1737 | Euler studies the zeta function |
| 1748 | Euler product formula explicitly connects primes and analysis |
| 1770 | Euler develops partition theory and continued fractions |
| Late 1700s | Early investigations of quadratic reciprocity |

## Nineteenth Century Foundations

| Period | Development |
|---|---|
| 1801 | entity["people","Carl Friedrich Gauss","German mathematician"] publishes Disquisitiones Arithmeticae |
| Early 1800s | Congruence notation becomes standard |
| 1829 | entity["people","Peter Gustav Lejeune Dirichlet","German mathematician"] proves infinitely many primes in arithmetic progressions |
| Mid 1800s | Algebraic number theory emerges |
| 1847 | entity["people","Ernst Kummer","German mathematician"] introduces ideal numbers |
| 1859 | entity["people","Bernhard Riemann","German mathematician"] publishes paper on zeta function |
| Late 1800s | entity["people","Richard Dedekind","German mathematician"] formalizes ideals |

## Early Twentieth Century

| Period | Development |
|---|---|
| 1896 | Prime Number Theorem proved independently by Hadamard and de la Vallée Poussin |
| Early 1900s | Class field theory develops |
| 1900 | entity["people","David Hilbert","German mathematician"] presents famous problems |
| 1920s | Local field theory becomes systematic |
| 1920s-1930s | Development of harmonic analysis and modular forms |
| 1930s | Modern algebra reshapes arithmetic foundations |

## Mid Twentieth Century

| Period | Development |
|---|---|
| 1940s | Weil conjectures formulated |
| 1950s | Adelic methods enter number theory |
| 1950s-1960s | Growth of algebraic geometry and cohomology |
| 1960s | entity["people","Robert Langlands","Canadian mathematician"] proposes Langlands program |
| 1960s | Modern automorphic representation theory develops |
| 1970s | Computational number theory accelerates |

## Late Twentieth Century

| Period | Development |
|---|---|
| 1977 | RSA cryptosystem introduced |
| 1980s | Elliptic curve cryptography proposed |
| 1980s | Modularity ideas connect elliptic curves and modular forms |
| 1994 | entity["people","Andrew Wiles","British mathematician"] proves Fermat’s Last Theorem |
| Late 1990s | Large computational databases become standard |

## Twenty-First Century

| Period | Development |
|---|---|
| Early 2000s | Rapid growth of arithmetic geometry and automorphic methods |
| 2000 | Riemann Hypothesis becomes a Clay Millennium Problem |
| 2000s | Large-scale computation of zeta zeros and modular forms |
| 2010s | Expansion of post-quantum cryptography research |
| 2010s | Advances in bounded prime gaps |
| 2020s | Increasing integration of computation, databases, and formal verification |

## Major Conceptual Transitions

| Era | Main Shift |
|---|---|
| Ancient arithmetic | concrete computation |
| Fermat and Euler | systematic arithmetic arguments |
| Gauss | structural congruence theory |
| Dirichlet and Riemann | analytic methods |
| Dedekind and Kummer | algebraic structures and ideals |
| Twentieth century | geometry, topology, and representations |
| Modern era | unification through Langlands and arithmetic geometry |

## Development of Major Subjects

| Subject | Approximate Emergence |
|---|---|
| Divisibility theory | Ancient Greece |
| Diophantine equations | Classical antiquity |
| Congruences | Early nineteenth century |
| Analytic number theory | Eighteenth and nineteenth centuries |
| Algebraic number theory | Nineteenth century |
| Local fields | Early twentieth century |
| Modular forms | Twentieth century |
| Elliptic curves | Twentieth century arithmetic formulation |
| Arithmetic geometry | Mid twentieth century |
| Computational number theory | Late twentieth century |
| Post-quantum cryptography | Twenty-first century |

## Long-Term Themes

Several ideas persist throughout the history of number theory:

| Theme | Historical Role |
|---|---|
| Prime numbers | structure of integers |
| Integer solutions | Diophantine problems |
| Symmetry | congruences and groups |
| Infinite processes | analytic methods |
| Geometry | arithmetic spaces |
| Local-global principles | field arithmetic |
| Computation | algorithms and cryptography |

The subject evolved from explicit calculation with integers into a broad theory connecting algebra, analysis, geometry, topology, computation, and representation theory.