Number theory, the queen of mathematics, studies integers and their properties. From the Chinese Remainder Theorem to Fermat's Last Theorem, this timeline covers key milestones from 300 CE to 2000 CE.
The Chinese mathematical text Sunzi Suanjing contains the earliest known statement of the Chinese Remainder Theorem, solving modular systems. #NumberTheory #HistoryOfMath
Aryabhata describes the Kuttaka ('pulverizer') algorithm for solving linear Diophantine equations. #NumberTheory #IndianMathematics
Brahmagupta's Brahmasphutasiddhanta provides the first explicit solution to Pell's equation, x^2 - Dy^2 = 1. #NumberTheory #IndianMathematics
Muhammad ibn Musa al-Khwarizmi writes 'The Compendious Book on Calculation by Completion and Balancing', introducing algebraic methods used in number theory. #NumberTheory #IslamicGoldenAge
Ibn al-Haytham (Alhazen) makes contributions to the theory of perfect numbers and congruences. #NumberTheory #IslamicScience
Leonardo of Pisa (Fibonacci) publishes Liber Abaci, introducing the Fibonacci sequence and promoting modular arithmetic in Europe. #NumberTheory #HistoryOfMath
Claude Gaspard Bachet de Méziriac publishes a Latin translation of Diophantus's Arithmetica, which later inspires Fermat. #NumberTheory #HistoryOfMath
Pierre de Fermat writes in the margin of his copy of Arithmetica that he has a proof that x^n + y^n = z^n has no integer solutions for n>2. #NumberTheory #Fermat
Fermat announces his little theorem: for prime p and integer a, a^p ≡ a (mod p). #NumberTheory #Fermat
Christian Goldbach conjectures that every even integer greater than 2 can be expressed as the sum of two primes. #NumberTheory #Conjecture
Leonhard Euler provides the first published proof of Fermat's little theorem and generalizes it. #NumberTheory #Euler
Euler introduces the totient function φ(n), counting numbers coprime to n. #NumberTheory #Euler
Joseph-Louis Lagrange proves that every natural number is the sum of at most four squares. #NumberTheory #Lagrange
Adrien-Marie Legendre conjectures an approximate formula for the distribution of primes, laying groundwork for the Prime Number Theorem. #NumberTheory #PrimeNumbers
Carl Friedrich Gauss publishes his magnum opus, establishing modern number theory. It covers modular arithmetic, quadratic reciprocity, and cyclotomic numbers. #NumberTheory #Gauss
Johann Peter Gustav Lejeune Dirichlet proves that there are infinitely many primes in any arithmetic progression a + nd with gcd(a,d)=1. #NumberTheory #Dirichlet
Évariste Galois develops the theory of finite fields (Galois fields), essential for number theory and cryptography. #NumberTheory #Galois
Dirichlet introduces L-functions to prove his theorem on primes in arithmetic progressions. #NumberTheory #Dirichlet
Ernst Kummer develops ideal numbers to salvage unique factorization in cyclotomic fields, aiming to prove Fermat's Last Theorem. #NumberTheory #Kummer
Pafnuty Chebyshev obtains bounds on the prime-counting function, giving the first significant result towards the Prime Number Theorem. #NumberTheory #Chebyshev
Bernhard Riemann publishes a paper on the distribution of primes, introducing the Riemann zeta function and stating the Riemann hypothesis. #NumberTheory #Riemann
Richard Dedekind introduces ideals in algebraic number theory, replacing Kummer's ideal numbers. #NumberTheory #Dedekind )
Charles Hermite proves that e, the base of natural logarithms, is transcendental, a major result in transcendentals. #NumberTheory #Transcendental
Ferdinand von Lindemann proves π is transcendental, confirming that squaring the circle is impossible. #NumberTheory #Transcendental
Jacques Hadamard and Charles de la Vallée-Poussin independently prove the Prime Number Theorem using complex analysis. #NumberTheory #PrimeNumbers
Kurt Hensel introduces p-adic numbers, a powerful tool in number theory. #NumberTheory #pAdic
David Hilbert presents 23 problems; among them are the Riemann hypothesis and Fermat's Last Theorem, shaping 20th-century number theory. #NumberTheory #Hilbert
David Hilbert proves that for every k, there exists a number g(k) such that every integer is a sum of g(k) k-th powers. #NumberTheory #Hilbert
Srinivasa Ramanujan sends a famous letter to G.H. Hardy containing over 100 theorems, starting a collaboration that produces major results in number theory. #NumberTheory #Ramanujan
G.H. Hardy and S. Ramanujan develop the circle method to obtain an asymptotic formula for the number of partitions. #NumberTheory #CircleMethod )
G.H. Hardy and John Edensor Littlewood refine the circle method for additive number theory, applying it to Goldbach problems. #NumberTheory #HardyLittlewood
Lev Schnirelmann proves that every integer can be expressed as sum of at most 300,000 primes, using density methods. #NumberTheory #Schnirelmann
Ivan Vinogradov proves that every sufficiently large odd integer is the sum of three primes. #NumberTheory #Vinogradov
Paul Erdős and Atle Selberg produce an elementary proof of the Prime Number Theorem, avoiding complex analysis. #NumberTheory #ErdosSelberg
Enrico Bombieri proves the large sieve and the Bombieri–Vinogradov theorem, a fundamental result in analytic number theory. #NumberTheory #Bombieri
Chen Jingrun proves that every sufficiently large even integer is the sum of a prime and a product of at most two primes. #NumberTheory #Chen
Alan Baker proves effective bounds for linear forms in logarithms, with applications to Diophantine equations. #NumberTheory #Baker
Pierre Deligne completes the proof of the Weil conjectures, a major triumph in algebraic number theory and geometry. #NumberTheory #WeilConjectures
Gerd Faltings proves that a curve of genus at least 2 has only finitely many rational points, a result that limits solutions to many Diophantine equations. #NumberTheory #Faltings
The Taniyama–Shimura conjecture (now modularity theorem) proposes a deep link between elliptic curves and modular forms. #NumberTheory #Modularity
Andrew Wiles begins a secret intensive effort to prove Fermat's Last Theorem, building on the Taniyama–Shimura conjecture. #NumberTheory #Fermat
Andrew Wiles announces a proof at a Cambridge conference, but a gap is later discovered. #NumberTheory #Fermat
Andrew Wiles and Richard Taylor fill the gap in the proof, finally proving Fermat's Last Theorem. #NumberTheory #Fermat
Wiles' corrected proof is published in the Annals of Mathematics, solving a 350-year-old problem. #NumberTheory #Fermat
The Clay Mathematics Institute includes the Riemann hypothesis as one of its seven Millennium Prize Problems, offering $1 million for a proof. #NumberTheory #MillenniumProblems