History of Number Theory Timeline

French mathematician Pierre de Fermat (1607–1665) never published his writings but communicated through correspondence and wrote in marginal notes instead. He founded modern number theory with results such as Fermat's little theorem, which was famously proved 358 years after its formulation as Fermat's Last Theorem.

Image source: Pierre de Fermat

Fermat wrote in a margin that he had found a proof too large for it to fit, claiming no positive integers satisfy xⁿ + yⁿ = zⁿ for n > 2. It remained unproven for 358 years, becoming one of the most famous open problems in mathematics.

Image source: Fermat's Last Theorem

The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur Christian Goldbach, pointed him towards some of Fermat's work on the subject. Goldbach's conjecture remains unsolved since the 18th century.

Image source: Christian Goldbach

Building on Fermat's work introduced to him by Goldbach, Euler proved many of Fermat's claims, generalized Fermat's little theorem, and made fundamental contributions to partitions, quadratic forms, and the distribution of primes.

Image source: Leonhard Euler

Goldbach conjectured in a letter to Euler that every even integer greater than two is the sum of two primes. The conjecture remains unsolved since the 18th century, despite extensive computational verification.

Image source: Goldbach's conjecture

Joseph-Louis Lagrange (1736–1813) gave full proofs of the four-square theorem, Wilson's theorem, and developed the basic theory of Pell's equations, putting many previously asserted results on rigorous footing.

Image source: Joseph-Louis Lagrange

Adrien-Marie Legendre (1752–1833) stated the law of quadratic reciprocity, describing when a prime is a quadratic residue modulo another prime. He also conjectured the prime number theorem.

Image source: Quadratic reciprocity

Gauss provided the first rigorous proof of the law of quadratic reciprocity at age nineteen and returned to it throughout his career, ultimately giving several distinct proofs that opened new avenues in number theory.

Image source: Carl Friedrich Gauss

Carl Friedrich Gauss (1777–1855) wrote Disquisitiones Arithmeticae (1801), which had an immense influence in the area of number theory and set its agenda for much of the 19th century. Gauss once remarked, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."

A conventional starting point for analytic number theory is Dirichlet's theorem on arithmetic progressions (1837), whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable.

Image source: Dirichlet's theorem on arithmetic progressions

Jacobi's four-square theorem (1839), which gives an exact formula for the number of representations of an integer as a sum of four squares, belongs to a strand of analytic number theory connected to modular forms that has now taken a leading role.

The use of complex analysis in number theory comes into its own with Bernhard Riemann's 1859 paper on the zeta function, the canonical starting point for analytic prime-counting methods. It introduced the Riemann hypothesis, still unresolved today.

Image source: Riemann hypothesis

The grounds of algebraic number theory were set in the late nineteenth century, when ideal numbers, the theory of ideals and valuation theory were introduced—three complementary ways of dealing with the lack of unique factorization in algebraic number fields.

The classification of abelian extensions of number fields was the object of the programme of class field theory, initiated in the late nineteenth century partly by Kronecker and Eisenstein and carried out largely between 1900 and 1950.

The prime number theorem, describing the asymptotic distribution of primes, was first proven using complex analysis in 1896 by Hadamard and de la Vallée Poussin, building on Riemann's zeta-function insights.

Image source: Prime number theorem

By the early twentieth century, the term 'number theory' had been widely adopted, replacing older names like 'arithmetic' for the discipline devoted to the integers and their generalizations.

Image source: Number theory

The number theorist Leonard Dickson (1874–1954) said "Thank God that number theory is unsullied by any application", reflecting the view of number theory as the epitome of pure mathematics before its cryptographic uses emerged.

For example, although the prime number theorem was first proven using complex analysis in 1896, an elementary proof avoiding complex analysis was found only in 1949 by Erdős and Selberg.

Number theory was long regarded as having no applications outside mathematics, until the 1970s, when prime numbers became the basis for the creation of public-key cryptography algorithms, such as the RSA cryptosystem.

Image source: RSA cryptosystem

In 1974, Donald Knuth said "virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations", marking number theory's growing role in computer science.

Image source: Donald Knuth