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| Carmichael number | |
|---|---|
| Name | Carmichael number |
| First known | 1910 |
| Discovered by | Robert Carmichael |
| Type | Composite integer with pseudoprime properties |
Carmichael number Carmichael numbers are composite integers that mimic prime behavior for many arithmetic tests; they satisfy Fermat-like congruences and therefore masquerade as primes in contexts involving Pierre de Fermat, Leonhard Euler, Évariste Galois, Carl Friedrich Gauss, and Adrien-Marie Legendre. Initially studied in the same era as work by Srinivasa Ramanujan, D. H. Lehmer, Paul Erdős, Graham Higman, and Alfréd Rényi, Carmichael numbers connect to research in cryptography, number theory, probability theory, analytic number theory, and computational projects led by institutions such as Bell Labs, University of Cambridge, Princeton University, Massachusetts Institute of Technology, and Institute for Advanced Study.
A Carmichael number is a composite integer n > 1 for which a^{n-1} ≡ 1 (mod n) holds for every integer a coprime to n, a property studied by Pierre de Fermat and extended by Joseph-Louis Lagrange, Adrien-Marie Legendre, Carl Friedrich Gauss, and Leonhard Euler. The formal modern criterion links to concepts investigated by Robert Carmichael and used in proofs by Alford, Granville and Pomerance and Korselt; it contrasts with primality properties analyzed by Miller–Rabin, Baillie–PSW, AKS primality test, and researchers from Bell Labs and IBM Research. Carmichael numbers are composite but satisfy Fermat-like congruences resembling those in work of Évariste Galois and Srinivasa Ramanujan.
The term and first examples trace to Robert Carmichael's 1910 note and earlier observations by Alfred Korselt in 1899; subsequent investigations involved Pafnuty Chebyshev's contemporaries and later proofs by Paul Erdős, R. D. Carmichael, Wacław Sierpiński, D. H. Lehmer, Atle Selberg, and G. H. Hardy. The problem attracted attention from scholars at University of Cambridge, Princeton University, Massachusetts Institute of Technology, University of Paris, and University of Göttingen and was advanced by research teams including Alford, Granville and Pomerance who proved infinitude using methods related to Turán, Vinogradov, I. M. Vinogradov, and analytic techniques similar to those in Yitang Zhang's and Terence Tao's work. Later computational verifications relied on projects affiliated with Project Euler, PrimePages, Los Alamos National Laboratory, and Oak Ridge National Laboratory.
Korselt's criterion, named for Alfred Korselt, states that a composite n is a Carmichael number if and only if n is squarefree and for every prime p dividing n, p − 1 divides n − 1; this characterization parallels divisibility concepts studied by Leonhard Euler and Carl Friedrich Gauss. Further properties tie to multiplicative order studies by Évariste Galois and to pseudoprime families examined by E. J. Lehmer, P. L. T. Nijhoff, and John Selfridge. Carmichael numbers are rare but infinite, a fact proved by W. R. Alford, Andrew Granville, and Carl Pomerance using sieve methods developed from theories of Atle Selberg and Paul Erdős. Their distribution relates to results by G. H. Hardy and Srinivasa Ramanujan on prime distribution and to conjectures influenced by C. P. Ramanujam and Rudolf Lidl.
Smallest examples include 561, 1105, 1729, and 2465, numbers that have been noted in computations by Robert Carmichael and later tabulated by projects at PrimePages and datasets maintained by University of Tennessee and Mathematica groups. The asymptotic density and counting function of Carmichael numbers were studied by Alford, Granville and Pomerance and refined by H. W. Lenstra and Pomerance with input from researchers at University of Georgia and University of Waterloo. Empirical surveys by teams at Los Alamos National Laboratory and Oak Ridge National Laboratory and computational work by Project Euler and PrimePages catalogued many examples, connecting to similar datasets from OEIS and software from Wolfram Research.
Because Carmichael numbers pass Fermat tests, stronger methods are required: deterministic tests like AKS primality test and probabilistic routines like Miller–Rabin primality test and Baillie–PSW primality test detect composites that Fermat tests miss. Practical identification uses factorization algorithms from John Pollard and Carl Pomerance (Pollard's rho, ECM) and implementations in software from Wolfram Research, PARI/GP, SageMath, GNU Multiple Precision Arithmetic Library, and code bases hosted by GitHub and computational groups at University of Cambridge and Princeton University. Research into zero-knowledge protocols at MIT Media Lab and Stanford University also considers Carmichael behavior in cryptographic contexts pioneered by Whitfield Diffie and Martin Hellman and developed by Ronald Rivest, Adi Shamir, and Leonard Adleman.
Generalizations include Korselt numbers, absolute pseudoprimes, strong Carmichael numbers, and elliptic Carmichael numbers; these relate to work by John Selfridge, Atkin, Morain, Joseph Silverman, Andrew Granville, and Carl Pomerance. Connections extend to pseudoprime research by R. D. Carmichael and analytic sieves from Atle Selberg and G. H. Hardy; elliptic analogues link to Andrew Wiles's and Richard Taylor's areas of study and to elliptic curve cryptography advanced by Neal Koblitz and Victor S. Miller. Broader themes intersect with studies by Paul Erdős, Terence Tao, Yitang Zhang, and Ben Green on arithmetic structure and distribution, and with algorithmic work from Bell Labs, IBM Research, and universities worldwide.
Category:Integer sequences