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A002997
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Carmichael numbers: composite numbers n such that a^(n-1) == 1 (mod n) for every a coprime to n.
(Formerly M5462)
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120
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561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, 488881, 512461
(list;
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refs;
listen;
history;
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internal format)
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OFFSET
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1,1
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COMMENTS
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An odd composite number n is a pseudoprime to base a iff a^(n-1) == 1 mod n. A Carmichael number is an odd composite number n which is a pseudoprime to base a for every number a prime to n.
Ghatage and Scott prove using Fermat's little theorem that (a+b)^n == a^n + b^n (mod n) (the freshman's dream) exactly when n is a prime (A000040) or a Carmichael number. - Jonathan Vos Post, Aug 31 2005
Alford et al. have constructed a Carmichael number with 10333229505 prime factors, and have also constructed Carmichael numbers with k prime factors for every k between 3 and 19565220. [Jonathan Vos Post, Apr 01 2012]
Thomas Wright proved that for any numbers b and M in N with gcd(b,M) = 1, there are infinitely many Carmichael numbers m such that m = b mod M. [Jonathan Vos Post, Dec 27 2012]
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REFERENCES
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F. Arnault, Constructing Carmichael numbers which are strong pseudoprimes to several bases, Journal of Symbolic Computation, vol. 20, no 2, Aug. 1995, pp. 151-161.
F. Arnault. Rabin-Miller primality test: Composite numbers which pass it, Mathematics of Computation, vol. 64, no 209, 1995, pp. 355-361.
F. Arnault. The Rabin-Monier theorem for Lucas pseudoprimes, Mathematics of Computation, vol. 66, no 218, April 1997, pp. 869-881.
A. H. Beiler, Recreations in the Theory of Numbers, Dover Publications, Inc. New York, 1966, Table 18, Page 44.
D. M. Burton, Elementary Number Theory, 5th ed., McGraw-Hill, 2002.
CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 87.
Pratibha Ghatage (p.ghatage(AT)csuohio.edu) and Brian Scott (b.scott(AT)csuohio.edu), When is (a+b)^n == a^n + b^n (mod n)?, College Mathematics Journal, Vol. 36, No. 4 (Sep 2005), p. 322.
R. K. Guy, Unsolved Problems in Number Theory, A13.
G. Jaeschke, The Carmichael numbers to 10^12, Math. Comp., 55 (1990), 383-389.
D. H. Lehmer, Errata for Poulet's table, Math. Comp., 25 (1971), 944-945.
O. Ore, Number Theory and Its History, McGraw-Hill, 1948, Reprinted by Dover Publications, 1988, Chapter 14.
P. Poulet, Tables des nombres composes verifiant le theoreme du Fermat pour le module 2 jusqu'a 100.000.000, Sphinx (Brussels), 8 (1938), 42-45.
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. - From N. J. A. Sloane, Feb 07 2013
W. Sierpi\'{n}ski, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 51.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (from the Pinch web site mentioned below)
W. R. Alford, Jon Grantham, Steven Hayman, Andrew Shallue, Constructing Carmichael numbers through improved subset-product algorithms, arXiv:1203.6664v1 [math.NT], Mar 29 2012.
W. R. Alford, A. Granville and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), no. 3, 703-722.
F. Arnault, Thesis
Joerg Arndt, Fxtbook, p. 786
J. Bernheiden, Carmichael numbers (Text in German)
C. K. Caldwell, The Prime Glossary, Carmichael number
Harvey Dubner, Journal of Integer Sequences, Vol. 5 (2002) Article 02.2.1, Carmichael Numbers of the form (6m+1)(12m+1)(18m+1).
A. Granville, Papers on Carmichael numbers
A. Granville, Primality testing and Carmichael numbers, Notices Amer. Math. Soc., 39 (No. 7, 1992), 696-700. [Broken link]
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Math. Comp. 71 (2002), no. 238, 883-908.
Renaud Lifchitz, A generalization of the Korselt's criterion - Nested Carmichael numbers [Broken link]
Yoshio Mimura, Carmichael Numbers up to 10^12
Math Reference Project, Carmichael Numbers
R. G. E. Pinch, Carmichael numbers up to 10^16 (FTP) [Broken link?]
R. G. E. Pinch, The Carmichael numbers up to 10^17
R. G. E. Pinch, Carmichael numbers up to 10^18, April 2006.
R. G. E. Pinch, The Carmichael numbers up to 10^18
F. Richman, Primality testing with Fermat's little theorem
Eric Weisstein's World of Mathematics, Carmichael Number
Eric Weisstein's World of Mathematics, Knoedel Numbers
Eric Weisstein's World of Mathematics, Pseudoprime
Wikipedia, Carmichael number
Thomas Wright, Infinitely Many Carmichael Numbers in Arithmetic Progressions, To appear in the Bulletin of the London Mathematical Society, arXiv:1212.5850v1 [math.NT].
Index entries for sequences related to Carmichael numbers.
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FORMULA
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n is composite and squarefree and for p prime, p|n => p-1|n-1.
A composite odd number n is a Carmichael number if and only if n is squarefree and p-1 divides n-1 for every prime p dividing n (Korselt, 1899)
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MATHEMATICA
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Cases[Range[1, 100000, 2], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]] (* Artur Jasinski, Apr 05 2008 *)
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PROG
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(PARI) Korselt(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1
isA002997(n)=n%2&!isprime(n)&Korselt(n)&n>1 \\ Charles R Greathouse IV, Jun 10 2011
(PARI) is_A002997(n)={my(f); bittest(n, 0) & !for(i=1, #f=factor(n)~, (f[2, i]==1 & n%(f[1, i]-1)==1)||return) & #f>1} \\ - M. F. Hasler, Aug 24 2012
(Haskell)
a002997 n = a002997_list !! (n-1)
a002997_list = [x | x <- a024556_list,
all (== 0) $ map ((mod (x - 1)) . (subtract 1)) $ a027748_row x]
-- Reinhard Zumkeller, Apr 12 2012
(MAGMA) [n: n in [3..53*10^4 by 2] | not IsPrime(n) and n mod CarmichaelLambda(n) eq 1]; // Bruno Berselli, Apr 23 2012
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CROSSREFS
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Cf. A001567, A064238-A064262, A006931, A055553, A002322, A083737, A153581, A024556, A027748.
Sequence in context: A006971 A218483 A104016 * A087788 A173703 A083733
Adjacent sequences: A002994 A002995 A002996 * A002998 A002999 A003000
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Replaced list of Carmichael numbers up to 10^9 by list up to 10^12. - Jan Kristian Haugland (admin(AT)neutreeko.net), Mar 25 2009
Minor edit of Mathematica code from Zak Seidov, Feb 16 2011
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STATUS
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approved
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