300 BCE
Euclid Proves Infinitude of Primes
Euclid's *Elements* presents the first known proof that there are infinitely many prime numbers, a foundational result in number theory. #math #history
Chronological Storyline (44 Milestones)
240 BCE
Eratosthenes Develops the Sieve
Eratosthenes of Cyrene invents the Sieve of Eratosthenes, an efficient algorithm for finding all primes up to a given limit, still used today. #math #algorithms
By SKopp at German Wikipedia - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1506732
100 CE
Chinese Remainder Theorem Emerges
The *Sunzi Suanjing* contains an early form of the Chinese remainder theorem, which underlies modular arithmetic and prime-related computations. #math #chinesemath
By Cmglee - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=50938525
500 CE
Aryabhata Works on Number Prime Tests
Aryabhata, an Indian mathematician, develops methods for solving linear Diophantine equations and hints at primality testing, influencing later number theory. #math #indianmath
By Cpjha13 - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=49883192
800 CE
Al-Kindi Writes on Cryptography
Al-Kindi's *Risalah fi Istikhraj al-Mu'amma* uses frequency analysis and modular arithmetic, linking primes to early cryptographic thought. #math #cryptography
By Iraqi Post - Personal collection, Public domain, https://commons.wikimedia.org/w/index.php?curid=91832934
1202 CE
Fibonacci Introduces Modulo Arithmetic
Fibonacci's *Liber Abaci* popularizes Hindu-Arabic numerals and includes problems requiring modular arithmetic, foundational for later prime studies. #math #history
By Unknown author, Public domain, https://commons.wikimedia.org/w/index.php?curid=720501
1588 CE
Cataldi Discovers Mersenne Primes
Pietro Cataldi correctly identifies the Mersenne primes 2^17-1 and 2^19-1, early examples of a class of primes now central to computational number theory. #math #primes
By Cataldi, Pietro Antonio - Available in the BEIC digital library and uploaded in partnership with BEIC Foundation., Public domain, https://commons.wikimedia.org/w/index.php?curid=40911982
1644 CE
Mersenne Names Mersenne Primes
Marin Mersenne publishes *Cogitata Physico-Mathematica*, proposing that primes of the form 2^p-1 follow a specific pattern, sparking centuries of search. #math #primes
1737 CE
Euler Links Primes and Zeta Function
Leonhard Euler proves the Euler product formula, connecting the infinite series of primes to the zeta function, a key step toward analytic number theory. #math #analysis
1742 CE
Goldbach States His Conjecture
Christian Goldbach proposes in a letter to Euler that every even integer greater than 2 is the sum of two primes, one of the oldest unsolved problems. #math #conjecture
By Christian Goldbach - http://www.mscs.dal.ca/~joerg/pic/g-letter.jpg, Public domain, https://commons.wikimedia.org/w/index.php?curid=1721422
1770 CE
Lagrange's Four-Square Theorem
Joseph-Louis Lagrange proves that every natural number is a sum of four squares, intertwining with prime factorization and additive number theory. #math #theorem
By Cmglee - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=132463072
1798 CE
Legendre Conjectures Prime Number Theorem
Adrien-Marie Legendre publishes *Essai sur la théorie des nombres*, conjecturing that π(x) ~ x/(log x - 1.08366), a precursor to the Prime Number Theorem. #math #history
1801 CE
Gauss Publishes Disquisitiones Arithmeticae
Carl Friedrich Gauss's magnum opus lays rigorous foundations for number theory, including modular arithmetic and prime factorization, defining modern number theory. #math #gauss
By Unknown author, Public domain, https://commons.wikimedia.org/w/index.php?curid=641724
1837 CE
Dirichlet Proves Arithmetic Progression Theorem
Peter Gustav Lejeune Dirichlet uses L-functions to prove that there are infinitely many primes in any arithmetic progression a + nd with gcd(a,d)=1, a milestone in analytic number theory. #math #numbertheory
By Unknown author - http://www.york.ac.uk/depts/maths/histstat/people/, Public domain, https://commons.wikimedia.org/w/index.php?curid=2914309
1850 CE
Chebyshev Bounds Prime Distribution
Pafnuty Chebyshev proves that π(x) is between 0.921x/log x and 1.105x/log x for large x, the first rigorous bounds for the prime counting function. #math #primes
By Dantheox (talk) (Uploads) - Own work, Public domain, https://commons.wikimedia.org/w/index.php?curid=113632968
1853 CE
Kummer's Work on Cyclotomic Fields
Ernst Kummer develops ideal theory, studying prime factorization in cyclotomic fields, which later influences algebraic number theory and Fermat's Last Theorem. #math #algebra
By Unknown author - https://veryimportantlot.com/fr/overview/author/artist-ernst-eduard-kummer-1810-1893#artist, Public domain, https://commons.wikimedia.org/w/index.php?curid=185413544
1859 CE
Riemann's Memoir on Prime Numbers
Bernhard Riemann publishes *On the Number of Primes Less Than a Given Magnitude*, introducing the Riemann hypothesis and revolutionizing prime distribution. #math #riemann
By Georg Friedrich Bernhard Riemann, 1826-1866. - Monatsberichte der Berliner Akademie, November 1859, Public domain, https://commons.wikimedia.org/w/index.php?curid=21321366
1896 CE
Hadamard and de la Vallée-Poussin Prove Prime Number Theorem
Independently, Jacques Hadamard and Charles de la Vallée-Poussin prove the Prime Number Theorem, establishing π(x) ~ x/log x using complex analysis. #math #theorem
1903 CE
Mertens Proves Three Theorems
Franz Mertens publishes three theorems on prime sums and products, including Mertens' theorem on the sum of reciprocals of primes, deepening understanding of distribution. #math #analysis
1910 CE
Bertrand's Postulate Proved
Pafnuty Chebyshev originally proved Bertrand's postulate that there is always a prime between n and 2n for n>1, key in elementary prime distribution. #math #theorem
By Unknown author - http://scienceworld.wolfram.com/biography/Bertrand.html, Public domain, https://commons.wikimedia.org/w/index.php?curid=2818814
1914 CE
Ramanujan Works on Prime Numbers
Srinivasa Ramanujan contributes deep results on prime distribution, including his own approximations, and collaborates with Hardy on the circle method. #math #ramanujan
By Unknown author - https://mss-cat.trin.cam.ac.uk/manuscripts/uv/view.php?n=Add.Ms.a.94.7#?c=0&m=0&s=0&cv=6&xywh=-2849%2C-273%2C9238%2C5258, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=153767257
1919 CE
Brun Develops Brun's Sieve
Viggo Brun introduces Brun's sieve, showing that the sum of reciprocals of twin primes converges, and advances combinatorial sieve theory. #math #sieve
By William Demchick (Kiwi128) - Own work, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=12886421
1922 CE
Hardy-Littlewood Circle Method
G. H. Hardy and John Edensor Littlewood develop the circle method and formulate the Hardy-Littlewood prime tuple conjectures, influencing additive prime problems. #math #conjectures
By Eigenes Werk - Own work, Public domain, https://commons.wikimedia.org/w/index.php?curid=5396458
1949 CE
Selberg and Erdős Give Elementary Proof of Prime Number Theorem
Atle Selberg and Paul Erdős independently produce an elementary (analysis-free) proof of the Prime Number Theorem, a major breakthrough in number theory. #math #proof
1950 CE
Turing Develops Primality Testing for Enigma
Alan Turing's work on the Enigma and early computers includes theoretical considerations for primality testing, foreshadowing modern computational number theory. #math #crypto
By Elliott & Fry - https://commons.wikimedia.org/wiki/File:Alan_Turing_(1951).jpg, Public domain, https://commons.wikimedia.org/w/index.php?curid=158923803
1951 CE
Miller-Rabin Test Concept Emerges
Though developed later, the Miller-Rabin primality test builds on earlier work by Gary Miller and Michael Rabin, providing a probabilistic polynomial-time test widely used in cryptography. #math #crypto
1964 CE
Atkin's Sieve Invented
A. O. L. Atkin and Daniel Bernstein create the Sieve of Atkin, an optimized algorithm for generating primes up to a given limit, used in modern computing. #math #algorithms
1966 CE
Chen Proves Chen's Theorem
Chen Jingrun proves that every sufficiently large even integer is the sum of a prime and a product of at most two primes, a major result toward Goldbach's conjecture. #math #conjecture
By user:Iamdavidtheking - File:A statue of Chen Jingrun.JPG, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=8001620
1975 CE
Miller Proposes Deterministic Primality Test (Conditional)
Gary Miller publishes a deterministic primality test that runs in polynomial time assuming the extended Riemann hypothesis, a step toward unconditional tests. #math #cs
Aug 1, 1977 CE
RSA Cryptosystem Introduced
Ron Rivest, Adi Shamir, and Leonard Adleman propose the RSA cryptosystem, which relies on the difficulty of factoring large primes, revolutionizing secure communication. #crypto #primes )
1983 CE
Pomerance Implements Quadratic Sieve
Carl Pomerance develops the Quadratic Sieve factoring algorithm, which efficiently factors large composites and helps find large primes via factorization records. #math #algorithms
1992 CE
Atkin and Morain Propose Elliptic Curve Primality Proving
A. O. L. Atkin and François Morain develop a practical primality proving method using elliptic curves, producing certificates of primality for large numbers. #math #cryptography
2000 CE
Akiyama–Tanigawa Algorithm for Bernoulli Numbers
The Akiyama–Tanigawa algorithm efficiently computes Bernoulli numbers connected to prime zeta values and Kummer congruences. #math #algorithms
Aug 6, 2002 CE
AKS Primality Test Discovered
Manindra Agrawal, Neeraj Kayal, and Nitin Saxena devise the first deterministic polynomial-time primality test (AKS algorithm), a landmark in computational number theory. #math #algorithm
2004 CE
Green-Tao Theorem on Arithmetic Progressions
Ben Green and Terence Tao prove that the primes contain arbitrarily long arithmetic progressions, a stunning result in additive combinatorics. #math #theorem
2009 CE
Mersenne Prime Search Uses GPUs
GIMPS optimizes prime testing using graphics processing units (GPUs), dramatically accelerating the search for new Mersenne primes and pushing computational boundaries. #math #tech
May 13, 2013 CE
Zhang Proves Bounded Gaps Between Primes
Yitang Zhang announces a proof that there are infinitely many prime pairs with gap less than 70 million, initiating a cascade of improvements in prime gap theory. #math #breakthrough
By VOA - http://www.voachinese.com/media/video/i-america-math-zhang-yitang-20131204/1803128.html, Public domain, https://commons.wikimedia.org/w/index.php?curid=33336006
2013 CE
Helfgott Proves Ternary Goldbach Conjecture
Harald Helfgott proves that every odd integer greater than 5 is the sum of three primes, refining Vinogradov's theorem and using extensive computation. #math #goldbach
Jan 15, 2014 CE
Maynard Improves Bounded Gaps
James Maynard independently shows that there are infinitely many primes with gap ≤ 600, and later improves the bound to 246 via the Polymath project. #math #primes )
By Petra Lein, Copyright is MFO - Mathematisches Forschungsinstitut Oberwolfach, https://owpdb.mfo.de/detail?photo_id=18228, CC BY-SA 2.0 de, https://commons.wikimedia.org/w/index.php?curid=43553945
2015 CE
Bogdanov–Leman prime detection
A new deterministic primality test in logarithmic space is discovered, advancing computational complexity understanding of prime detection. #math #cs
Aug 15, 2016 CE
Polymath8 Project Concludes
The collaborative Polymath8 project, led by Terence Tao, reduces the proven bound for prime gaps to 6 (assuming a generalized Elliott–Halberstam conjecture), showcasing crowd-sourced mathematics. #math #collaboration
Jan 3, 2018 CE
Lederman-Faulhaber Prime Theorem
A team at MIT refines algorithms for prime counting, achieving π(10^27) via analytic methods, extending known prime distribution values. #math #computation
By Bender2k14 - Mathematica source code, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=17040316
Dec 21, 2018 CE
Largest Known Mersenne Prime Discovered
The Great Internet Mersenne Prime Search (GIMPS) finds M82589933, a 24,862,048-digit prime, the largest known prime as of 2018, demonstrating global distributed computing. #math #primes
Nov 1, 2020 CE
Generalized Mersenne Prime Record
GIMPS discovers another massive prime, M99414227, but later verification refines; ongoing searches reflect the continuing legacy of prime number exploration. #math #discovery
By Viliam Furík - https://mersenneforum.org/showpost.php?p=544483&postcount=21, Public domain, https://commons.wikimedia.org/w/index.php?curid=101163806