Chronology of Number Theory – brain

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.