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.