Timeline of number theory

A timeline of number theory.

Before 1000 BCE

• ca.
  Egyptian multiplication although this is disputed.^[1]

• 300 BCE — Euclid proves the number of prime numbers is infinite.

1st millennium AD

• 250 — Diophantus writes Arithmetica, one of the earliest treatises on algebra.
• 500 — Aryabhata solves the general linear diophantine equation.
• 628 - Brahmagupta gives Brahmagupta's identity and solves the so called Pell's equation using his composition method.
• ca. 650 — Mathematicians in India create the Hindu–Arabic numeral system we use, including the zero, the decimals and negative numbers.

1000–1500

• ca. 1000 —
  Abu-Mahmud al-Khujandi first states a special case of Fermat's Last Theorem
  .
• 895 —
  proper divisors
  of the other).
• 975 — The earliest triangle of binomial coefficients (Pascal triangle) occur in the 10th century in commentaries on the Chandas Shastra.
• 1150 —
  Bhaskara II gives first general method for solving Pell's equation
• 1260 —
  Fermat as well as Thabit ibn Qurra.^[2]

• 1637 — Pierre de Fermat claims to have proven Fermat's Last Theorem in his copy of Diophantus' Arithmetica.

18th century

• 1742 — Christian Goldbach conjectures that every even number greater than two can be expressed as the sum of two primes, now known as Goldbach's conjecture.
• 1770 —
  Joseph Louis Lagrange proves the four-square theorem, that every positive integer is the sum of four squares of integers. In the same year, Edward Waring conjectures Waring's problem
  , that for any positive integer k, every positive integer is the sum of a fixed number of k^th powers.
• 1796 — Adrien-Marie Legendre conjectures the prime number theorem.

19th century

• 1801 — Disquisitiones Arithmeticae, Carl Friedrich Gauss's number theory treatise, is published in Latin.
• 1825 — Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre prove Fermat's Last Theorem for n = 5.
• 1832 — Lejeune Dirichlet proves Fermat's Last Theorem for n = 14.
• 1835 — Lejeune Dirichlet proves
  arithmetic progressions
  .
• 1859 — Bernhard Riemann formulates the Riemann hypothesis which has strong implications about the distribution of prime numbers.
• 1896 —
  Charles Jean de la Vallée-Poussin independently prove the prime number theorem
  .
• 1896 — Hermann Minkowski presents Geometry of numbers.

20th century

• 1903 —
  Edmund Georg Hermann Landau
  gives considerably simpler proof of the prime number theorem.
• 1909 — David Hilbert proves Waring's problem.
• 1912 — Josip Plemelj publishes simplified proof for the Fermat's Last Theorem for exponent n = 5.
• 1913 —
  Srinivasa Aaiyangar Ramanujan sends a long list of complex theorems without proofs to G. H. Hardy
  .
• 1914 — Srinivasa Aaiyangar Ramanujan publishes Modular Equations and Approximations to π.
• 1910s — Srinivasa Aaiyangar Ramanujan develops over 3000 theorems, including properties of
  hypergeometric series
  and prime number theory.
• 1919 —
  Brun's constant B₂ for twin primes
  .
• 1937 —
  I. M. Vinogradov proves Vinogradov's theorem that every sufficiently large odd integer is the sum of three primes, a close approach to proving Goldbach's weak conjecture
  .
• 1949 — Atle Selberg and Paul Erdős give the first elementary proof of the prime number theorem.
• 1966 —
  Goldbach conjecture
  .
• 1967 — Robert Langlands formulates the influential Langlands program of conjectures relating number theory and representation theory.
• 1983 —
  Mordell conjecture
  and thereby shows that there are only finitely many whole number solutions for each exponent of Fermat's Last Theorem.
• 1994 —
  Taniyama–Shimura conjecture and thereby proves Fermat's Last Theorem
  .
• 1999 — the full Taniyama–Shimura conjecture is proved.

• 2002 —
  polynomial time algorithm to determine whether a given number is prime
  .
• 2002 — Preda Mihăilescu proves Catalan's conjecture.
• 2004 —
  Ben Green and Terence Tao prove the Green–Tao theorem, which states that the sequence
  of prime numbers contains arbitrarily long arithmetic progressions.