Timeline of number theory

Last updated July 11, 2022

Before 1000 BCE

ca. 20,000 BCE — Nile Valley, Ishango Bone: possibly the earliest reference to prime numbers and Egyptian multiplication although this is disputed.^[1]

About 300 BCE

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.
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 — Thabit ibn Qurra gives a theorem by which pairs of amicable numbers can be found, (i.e., two numbers such that each is the sum of the 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 — Al-Farisi gave a new proof of Thābit ibn Qurra's theorem, introducing important new ideas concerning factorization and combinatorial methods. He also gave the pair of amicable numbers 17296 and 18416 which have also been jointly attributed to Fermat as well as Thabit ibn Qurra.^[2]

17th century

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 Dirichlet's theorem about prime numbers in arithmetic progressions.
1859 — Bernhard Riemann formulates the Riemann hypothesis which has strong implications about the distribution of prime numbers.
1896 — Jacques Hadamard and 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 highly composite numbers, the partition function and its asymptotics, and mock theta functions. He also makes major breakthroughs and discoveries in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory.
1919 — Viggo Brun defines 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 — Chen Jingrun proves Chen's theorem, a close approach to proving the Goldbach conjecture.
1967 — Robert Langlands formulates the influential Langlands program of conjectures relating number theory and representation theory.
1983 — Gerd Faltings proves the Mordell conjecture and thereby shows that there are only finitely many whole number solutions for each exponent of Fermat's Last Theorem.
1994 — Andrew Wiles proves part of the Taniyama–Shimura conjecture and thereby proves Fermat's Last Theorem.
1999 — the full Taniyama–Shimura conjecture is proved.

21st century

2002 — Manindra Agrawal, Nitin Saxena, and Neeraj Kayal of IIT Kanpur present an unconditional deterministic 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.

Related Research Articles

Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, σ(a)=b+a and σ(b)=a+b, where σ(n) is equal to the sum of positive divisors of n.

Conjecture Proposition in mathematics that is unproven

In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's Last Theorem, have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers or defined as generalizations of the integers.

In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers raised to the power k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants".

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than two is the sum of two prime numbers.

In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that

The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers and additive number theory.

Ribet's theorem is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof was a significant step towards the proof of Fermat's Last Theorem (FLT). As shown by Serre and Ribet, the Taniyama–Shimura conjecture and the epsilon conjecture together imply that FLT is true.

Vorlesungen über Zahlentheorie is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Others were written by Leopold Kronecker, Edmund Landau, and Helmut Hasse. They all cover elementary number theory, Dirichlet's theorem, quadratic fields and forms, and sometimes more advanced topics.

In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as:

A modular elliptic curve is an elliptic curve E that admits a parametrisation X₀(N) → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular.

In number theory, Vinogradov's theorem is a result which implies that any sufficiently large odd integer can be written as a sum of three prime numbers. It is a weaker form of Goldbach's weak conjecture, which would imply the existence of such a representation for all odd integers greater than five. It is named after Ivan Matveyevich Vinogradov who proved it in the 1930s. Hardy and Littlewood had shown earlier that this result followed from the generalized Riemann hypothesis, and Vinogradov was able to remove this assumption. The full statement of Vinogradov's theorem gives asymptotic bounds on the number of representations of an odd integer as a sum of three primes. The notion of "sufficiently large" was ill-defined in Vinogradov's original work, but in 2002 it was shown that 10¹³⁴⁶ is sufficiently large. Additionally numbers up to 10²⁰ had been checked via brute force methods, thus only a finite number of cases to check remained before the odd Goldbach conjecture would be proven or disproven. In 2013, Harald Helfgott proved Goldbach's weak conjecture for all cases.

In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve

Fermats Last Theorem 17th century conjecture proved by Andrew Wiles in 1994

In number theory, Fermat's Last Theorem states that no three positive integers $a$ , $b$ , and $c$ satisfy the equation $a n + b n = c n$ for any integer value of $n$ greater than 2. The cases $n = 1$ and $n = 2$ have been known since antiquity to have infinitely many solutions.

This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.

Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were almost universally considered inaccessible to prove by contemporaneous mathematicians, meaning that they were believed to be impossible to prove using current knowledge.

Fermat's Last Theorem is a theorem in number theory, originally stated by Pierre de Fermat in 1637 and proved by Andrew Wiles in 1995. The statement of the theorem involves an integer exponent n larger than 2. In the centuries following the initial statement of the result and before its general proof, various proofs were devised for particular values of the exponent n. Several of these proofs are described below, including Fermat's proof in the case n = 4, which is an early example of the method of infinite descent.

References

↑ Rudman, Peter Strom (2007). How Mathematics Happened: The First 50,000 Years. Prometheus Books. p. 64. ISBN 978-1-59102-477-4.
↑ Various AP Lists and Statistics Archived 2012-07-28 at the Wayback Machine

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[1] Rudman, Peter Strom (2007). How Mathematics Happened: The First 50,000 Years. Prometheus Books. p. 64. ISBN 978-1-59102-477-4.

[2] Various AP Lists and Statistics Archived 2012-07-28 at the Wayback Machine

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