300 BCE
Euclid's Proof of Infinitude of Primes
Euclid's 'Elements' includes a proof that there are infinitely many prime numbers, a foundational result in number theory. #mathematics #history
Chronological Storyline (42 Milestones)
200 BCE
Sieve of Eratosthenes
Eratosthenes of Cyrene devises an algorithm to find all prime numbers up to a given limit, known as the Sieve of Eratosthenes. #mathematics #algorithm
By SKopp at German Wikipedia - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1506732
300 CE
Chinese Remainder Theorem
Sunzi Suanjing describes a method for solving simultaneous congruences, now known as the Chinese remainder theorem, a key tool in prime factorization. #mathematics #China
By Cmglee - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=50938525
1640 CE
Fermat's Little Theorem
Pierre de Fermat states that if p is a prime and a is not divisible by p, then a^{p-1} ≡ 1 mod p, a fundamental tool in primality testing. #mathematics #primes
1644 CE
Mersenne Primes Introduced
Marin Mersenne publishes 'Cogitata Physica-Mathematica', discussing numbers of the form 2^p - 1, later known as Mersenne primes, which are central to prime searches. #mathematics #primes
1650 CE
Fermat Primes Conjectured
Pierre de Fermat conjectures that numbers of the form 2^{2^n}+1 are prime; later disproved for n=5, but they remain a key concept in number theory. #mathematics #primes
1737 CE
Euler's Product Formula
Leonhard Euler introduces the product formula linking the Riemann zeta function to primes, providing a new approach to prime distribution. #mathematics #zeta
1742 CE
Goldbach's Conjecture
Christian Goldbach proposes in a letter to Euler that every even integer greater than 2 can be expressed as the sum of two primes, an unsolved problem. #mathematics #primes
By Christian Goldbach - http://www.mscs.dal.ca/~joerg/pic/g-letter.jpg, Public domain, https://commons.wikimedia.org/w/index.php?curid=1721422
1792 CE
Gauss's Conjecture on Prime Distribution
At age 15, Carl Friedrich Gauss conjectures that the prime-counting function π(x) is asymptotically x/ln(x), later known as the Prime Number Theorem. #mathematics #history
1798 CE
Legendre's Conjecture
Adrien-Marie Legendre publishes 'Essai sur la Théorie des Nombres', stating that π(x) ≈ x/(ln(x)-1.08366), and later conjecturing there is always a prime between n^2 and (n+1)^2. #mathematics #primes
1801 CE
Gauss's Disquisitiones Arithmeticae
Gauss's monumental work systematically develops number theory, including quadratic reciprocity and foundational results on primes. #mathematics #history
By Unknown author, Public domain, https://commons.wikimedia.org/w/index.php?curid=641724
1825 CE
Sophie Germain's Work on Primes
Sophie Germain shows that for Fermat's Last Theorem, primes of the form 2p+1 (Sophie Germain primes) are significant, advancing prime theory. #mathematics #women
1837 CE
Dirichlet's Theorem on Arithmetic Progressions
Peter Gustav Lejeune Dirichlet proves that any arithmetic progression a, a+d, a+2d,... with gcd(a,d)=1 contains infinitely many primes. #mathematics #primes
1852 CE
Chebyshev's Theorem on Prime Distribution
Pafnuty Chebyshev proves bounds for π(x) showing that π(x) is between 0.92 x/ln(x) and 1.11 x/ln(x) for sufficiently large x. #mathematics #primes
By Dantheox (talk) (Uploads) - Own work, Public domain, https://commons.wikimedia.org/w/index.php?curid=113632968
1859 CE
Riemann Hypothesis Proposed
Bernhard Riemann's paper 'On the Number of Primes Less Than a Given Magnitude' introduces the Riemann zeta function and the hypothesis about its zeros, a central unsolved problem. #mathematics #primes
By No machine-readable author provided. Conscious assumed (based on copyright claims). - No machine-readable source provided. Own work assumed (based on copyright claims)., CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=939039
1876 CE
Lucas-Lehmer Primality Test
Édouard Lucas develops a test for Mersenne primes, later refined by Derrick Lehmer; it remains the most efficient test for large primes. #mathematics #primes
By KurtSchwitters - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=11675028
1896 CE
Prime Number Theorem Proved
Jacques Hadamard and Charles de la Vallée-Poussin independently prove the Prime Number Theorem, showing π(x) ~ x/ln(x). #mathematics #history
1900 CE
Dickson's Conjecture
Leonard Eugene Dickson proposes a general conjecture on prime k-tuples, which includes the twin prime conjecture as a special case. #mathematics #primes
1912 CE
Carmichael Numbers Discovered
Robert Carmichael finds composite numbers that satisfy Fermat's little theorem, known as Carmichael numbers, highlighting limitations of simple primality tests. #mathematics #primes
By no coneguts - U-Divulga - Universitat de Vic: https://mon.uvic.cat/udivulga/efemerides/marc/robert-daniel-carmichael/, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=87459394
1913 CE
Ramanujan's Contributions to Prime Distribution
Srinivasa Ramanujan publishes work on the prime-counting function and the Ramanujan prime, advancing the understanding of prime distribution. #mathematics #India
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
1914 CE
Littlewood's Theorem on π(x) - li(x)
John Edensor Littlewood proves that the difference π(x) - li(x) changes sign infinitely often, overturning earlier beliefs. #mathematics #primes
1919 CE
Brun's Theorem on Twin Primes
Viggo Brun shows that the sum of reciprocals of twin primes converges, a milestone in the study of prime gaps. #mathematics #primes
By William Demchick (Kiwi128) - Own work, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=12886421
1923 CE
Hardy-Littlewood Circle Method and Primes
G. H. Hardy and John Littlewood develop the circle method and formulate conjectures about prime k-tuples, deeply influencing analytic number theory. #mathematics #primes
1925 CE
Lehmer's Primality Test
Derrick Henry Lehmer devises a deterministic primality test based on Lucas sequences, a precursor to modern primality tests. #mathematics #primes
1936 CE
Cramér's Conjecture
Harald Cramér proposes that prime gaps are O(log^2 p), a heuristic law for the distribution of prime gaps. #mathematics #primes
1940 CE
Erdős–Kac Theorem
Paul Erdős and Mark Kac prove that the number of prime factors of a large integer has a normal distribution, a landmark in probabilistic number theory. #mathematics #primes
1952 CE
First Computer-Discovered Mersenne Prime
Using the SWAC computer, Derrick Lehmer and Raphael Robinson discover Mersenne prime M521, the first prime found with a computer. #mathematics #computing
1958 CE
Polya's Conjecture Disproved
C. B. Haselgrove disproves Polya's conjecture that most numbers have an odd number of prime factors, using a computer search. #mathematics #primes
By Linas Vepstas (User:Linas) - w:File:Lioville-big.svg, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1869174
1963 CE
Ulam Spiral Discovered
Stanislaw Ulam discovers the Ulam spiral, a graphical representation that reveals unexpected diagonal patterns in prime distribution. #mathematics #primes
By Morn - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=184101362
1966 CE
Chen's Theorem
Jingrun Chen proves that every sufficiently large even number is the sum of a prime and a product of at most two primes, a major step toward Goldbach. #mathematics #China
By user:Iamdavidtheking - File:A statue of Chen Jingrun.JPG, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=8001620
1976 CE
Miller-Rabin Primality Test
Gary Miller and Michael Rabin develop a probabilistic primality test based on Fermat's theorem and strong pseudoprimes, widely used today. #mathematics #algorithms
1980 CE
Baillie-PSW Primality Test
Robert Baillie, Carl Pomerance, and Samuel Wagstaff devise a combined test (Lucas and Fermat) that has no known counterexamples. #mathematics #primes
1983 CE
APR Primality Test
Leonard Adleman, Carl Pomerance, and Robert Rumely introduce a deterministic primality test using cyclotomic fields, practical for moderate numbers. #mathematics #primes
1986 CE
Elliptic Curve Primality Proving
Shafi Goldwasser and Joe Kilian develop an algorithm to prove primality using elliptic curves, offering an efficient certificate-based method. #mathematics #cryptography
1996 CE
Great Internet Mersenne Prime Search (GIMPS) Founded
George Woltman launches GIMPS, a distributed computing project to find Mersenne primes, discovering the largest known primes. #mathematics #computing
By Viliam Furík - https://mersenneforum.org/showpost.php?p=544483&postcount=21, Public domain, https://commons.wikimedia.org/w/index.php?curid=101163806
2002 CE
AKS Primality Test Discovered
Manindra Agrawal, Neeraj Kayal, and Nitin Saxena announce the first deterministic polynomial-time primality test, a landmark in computational number theory. #mathematics #algorithms
2004 CE
Green–Tao Theorem on Arithmetic Progressions of Primes
Ben Green and Terence Tao prove that the primes contain arbitrarily long arithmetic progressions, a stunning result in additive combinatorics. #mathematics #primes
2013 CE
Zhang's Bounded Gaps Between Primes
Yitang Zhang proves that there are infinitely many prime pairs with gap less than 70 million, a breakthrough in the twin prime problem. #mathematics #primes
2013 CE
Helfgott Proves Ternary Goldbach Conjecture
Harald Helfgott proves the weak Goldbach conjecture: every odd number greater than 5 is the sum of three primes. #mathematics #primes
2014 CE
Maynard and Tao Improve Bounded Gaps
James Maynard and Terence Tao independently refine Zhang's result, showing infinitely many prime gaps no larger than 600 (later 246). #mathematics #primes
2016 CE
Polymath Project on Prime Gaps
The Polymath 8b project, led by Terence Tao, reduces the proven bound for gaps to 6 (subject to a generalized Elliott-Halberstam conjecture). #mathematics #collaboration
2018 CE
Largest Known Prime (M82589933) Discovered
GIMPS discovers M82589933, a Mersenne prime with 24,862,048 digits, the largest known prime as of 2023. #mathematics #computing