Prime (mathematics)
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| Prime (mathematics) | |
|---|---|
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| Name | Prime (mathematics) |
| Field | Number theory |
| Introduced | Antiquity |
| Notable | Euclid, Eratosthenes, Euler, Gauss, Riemann |
Prime (mathematics) A prime is a positive integer greater than 1 that has no positive divisors other than 1 and itself. Primes form the multiplicative atoms of the integers and underpin results in Euclid, Eratosthenes, Leonhard Euler, Carl Friedrich Gauss, and Bernhard Riemann. Their study links to conjectures associated with Pierre de Fermat, Srinivasa Ramanujan, John von Neumann, Andrew Wiles, and modern institutions such as the Princeton University, Cambridge University, and Institute for Advanced Study.
Definition and basic properties
A prime is an element of the ring of integers with exactly two distinct positive divisors; composite numbers factor into primes uniquely by the Fundamental Theorem of Arithmetic proved in texts associated with Euclid and elaborated by Gauss. Basic properties include divisibility relations discovered in works by Diophantus and exploited by Eratosthenes for sieve methods; primes greater than 2 are odd, and except for 2 and 3 all primes lie in residue classes described in studies linked to Pierre de Fermat and Joseph-Louis Lagrange. Multiplicative structure, unit theory, and unique factorization relate to developments at Royal Society meetings and to algebraic number theory founded by Ernst Kummer, Richard Dedekind, and Heinrich Weber.
Distribution of primes
The distribution of primes across the integers is governed asymptotically by the Prime Number Theorem proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin, and anticipated in writings by Gauss and Adrien-Marie Legendre. Error bounds and zero-free regions for zeta functions connect to the Riemann zeta function studied by Bernhard Riemann and later to the Generalized Riemann Hypothesis considered by David Hilbert and Alan Turing. Empirical patterns and gaps between primes motivated conjectures by Polignac and G.H. Hardy with later breakthroughs at institutions such as Massachusetts Institute of Technology and University of Oxford. Statistical models of primes draw on methods from analysts like Atle Selberg and probabilists such as Paul Erdős and led to the study of primes in arithmetic progressions in work by Dirichlet, culminating in results connected to Hecke and Artin.
Primality testing and factorization
Primality testing algorithms progressed from deterministic checks in treatises by Euclid to modern algorithms like the AKS primality test developed at Princeton University and randomized algorithms influenced by Manuel Blum and Michael Rabin. Public-key cryptography protocols such as RSA (cryptosystem) and schemes arising from research at Bell Laboratories make heavy use of large primes and depend on the difficulty of integer factorization, a problem addressed by algorithms including the quadratic sieve from researchers at CWI and the general number field sieve developed by collaborations involving Hendrik Lenstra and John Pollard. Quantum algorithms proposed by Peter Shor at AT&T Bell Labs threaten classical factoring assumptions, prompting investigations at IBM and Microsoft Research.
Arithmetic functions and generalizations
Arithmetic functions tied to primes include the Möbius function, the Liouville function, and the von Mangoldt function, all central in analytic work by Srinivasa Ramanujan and G.H. Hardy and in explicit formulas due to Riemann and later expositions at Collège de France. Multiplicative functions and Dirichlet convolutions underpin the theory of L-functions explored by Erich Hecke and Haruzo Hida, while generalizations of primes arise in algebraic number fields and Dedekind domains studied by Richard Dedekind and Emmy Noether. Concepts such as prime ideals, irreducible elements, and units connect to research at University of Göttingen and later to modern algebraic geometry influenced by Alexander Grothendieck.
History and notable problems in prime research
Historical milestones include ancient listings and sieves credited to Eratosthenes, Euclid’s proof of infinitude referenced in works circulated at the Library of Alexandria, and 19th–20th century advances by Gauss, Dirichlet, Riemann, Hardy, and Littlewood. Landmark unresolved problems and achievements involve the Riemann Hypothesis proposed by Bernhard Riemann; the twin prime conjecture investigated by Alphonse de Polignac and advanced by recent results from researchers connected to University of Oxford and Yitang Zhang; Goldbach-type statements dating to Christian Goldbach and studied by Euler; and Catalan’s conjecture resolved by Péter Mihăilescu. Major computational projects led by teams at University of Tennessee, University of California, Berkeley, and National Institute of Standards and Technology have verified properties of large primes and Mersenne primes catalogued by the Great Internet Mersenne Prime Search and enthusiasts associated with GIMPS. Ongoing research spans analytic, algebraic, and computational directions informed by collaborations at Clay Mathematics Institute, Simons Foundation, and international conferences such as those sponsored by the American Mathematical Society.