Search: id:A002997
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%I A002997 M5462
%S A002997 561,1105,1729,2465,2821,6601,8911,10585,15841,29341,41041,46657,52633,
%T A002997 62745,63973,75361,101101,115921,126217,162401,172081,188461,252601,
%U A002997 278545,294409,314821,334153,340561,399001,410041,449065,488881,512461
%N A002997 Carmichael numbers: composite numbers n such that a^{n-1} = 1 ( mod n) 
               if a is prime to n.
%C A002997 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.
%C A002997 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 (jvospost3(AT)gmail.com), 
               Aug 31 2005
%D A002997 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002997 Alford, W. R., Granville, Andrew and Pomerance, Carl, There are infinitely 
               many Carmichael numbers, Ann. of Math. (2) 139 (1994), no. 3, 703-722.
%D A002997 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.
%D A002997 F. Arnault. Rabin-Miller primality test: Composite numbers which pass 
               it, Mathematics of Computation, vol. 64, no 209, 1995, pp. 355-361.
%D A002997 F. Arnault. The Rabin-Monier theorem for Lucas pseudoprimes, Mathematics 
               of Computation, vol. 66, no 218, April 1997, pp. 869-881.
%D A002997 A. H. Beiler, Recreations in the Theory of Numbers, Dover Publications, 
               Inc. New York, 1966, Table 18, Page 44.
%D A002997 D. M. Burton, Elementary Number Theory, 5th ed., McGraw-Hill, 2002.
%D A002997 CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 87.
%D A002997 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.
%D A002997 Granville, Andrew and Pomerance, Carl, Two contradictory conjectures 
               concerning Carmichael numbers. Math. Comp. 71 (2002), no. 238, 883-908.
%D A002997 R. K. Guy, Unsolved Problems in Number Theory, A13.
%D A002997 G. Jaeschke, The Carmichael numbers to 10^12, Math. Comp., 55 (1990), 
               383-389.
%D A002997 D. H. Lehmer, Errata for Poulet's table, Math. Comp., 25 (1971), 944-945.
%D A002997 O. Ore, Number Theory and Its History, McGraw-Hill, 1948, Reprinted by 
               Dover Publications, 1988, Chapter 14.
%D A002997 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.
%D A002997 W. Sierpi\'{n}ski, A Selection of Problems in the Theory of Numbers. 
               Macmillan, NY, 1964, p. 51.
%H A002997 N. J. A. Sloane, <a href="b002997.txt">Table of n, a(n) for n = 1..10000</
               a> (from the Pinch web site mentioned below)
%H A002997 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A002997 W. R. Alford, A. Granville and C. Pomerance, <a href="http://www.dms.umontreal.ca/
               ~andrew/Postscript/carmichael.ps">There are infinitely many Carmichael 
               numbers</a>, Ann. of Math. (2) 139 (1994), no. 3, 703-722.
%H A002997 F. Arnault, <a href="http://www.unilim.fr/~laco/perso/francois.arnault/
               these.ps">Thesis</a>
%H A002997 J. Bernheiden, <a href="http://www.mathe-schule.de/download/pdf/Primzahl/
               Carmichaelzahlen.pdf">Carmichael numbers (Text in German)</a>
%H A002997 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
               page.php?sort=CarmichaelNumber">Carmichael number</a>
%H A002997 Harvey Dubner, Journal of Integer Sequences, Vol. 5 (2002) Article 02.2.1, 
               <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Dubner/dubner6.html">
               Carmichael Numbers of the form (6m+1)(12m+1)(18m+1).</a>
%H A002997 A. Granville, <a href="http://www.dms.umontreal.ca/~andrew/agpapers.html">
               Papers on Carmichael numbers</a>
%H A002997 A. Granville, <a href="http://www.dms.umontreal.ca/~andrew/Postscript/
               notices.ps">Primality testing and Carmichael numbers</a>, Notices 
               Amer. Math. Soc., 39 (No. 7, 1992), 696-700.
%H A002997 Renaud Lifchitz, <a href="http://ourworld.compuserve.com/homepages/hlifchitz/
               Renaud.html">A generalization of the Korselt's criterion - Nested 
               Carmichael numbers</a>
%H A002997 Yoshio Mimura, <a href="http://www.kobepharma-u.ac.jp/~math/notes/note02.html">
               Carmichael Numbers up to 10^12</a>
%H A002997 Math Reference Project, <a href="http://www.mathreference.com/num-mod,
               ccm.html">Carmichael Numbers</a>
%H A002997 R. G. E. Pinch, <a href="ftp://ftp.dpmms.cam.ac.uk/pub/Carmichael/">Carmichael 
               numbers up to 10^16 (FTP)</a>
%H A002997 R. G. E. Pinch, <a href="http://arXiv.org/abs/math.NT/0504119">The Carmichael 
               numbers up to 10^17</a>
%H A002997 Richard Pinch, <a href="http://www.chalcedon.demon.co.uk/rgep/publish.html#76">
               Carmichael numbers up to 10^18</a>, April 2006.
%H A002997 R. G. E. Pinch, <a href="http://arXiv.org/abs/math/0604376">The Carmichael 
               numbers up to 10^18</a>
%H A002997 F. Richman, <a href="http://www.math.fau.edu/Richman/carm.htm">Primality 
               testing with Fermat's little theorem</a>
%H A002997 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CarmichaelNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A002997 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               KnoedelNumbers.html">Link to a section of The World of Mathematics.</
               a>
%H A002997 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Pseudoprime.html">Link to a section of The World of Mathematics.</
               a>
%H A002997 Wikipedia, <a href="http://www.wikipedia.org/wiki/Carmichael_number">
               Carmichael number</a>
%H A002997 <a href="Sindx_Ca.html#Carmichael">Index entries for sequences related 
               to Carmichael numbers.</a>
%F A002997 n is composite and square-free and for p prime, p|n => p-1|n-1.
%F A002997 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)
%t A002997 Cases[Range[100000], n_ /; Mod[n, CarmichaelLambda[n]] == 1 && ! PrimeQ[n]] 
               - Artur Jasinski (grafix(AT)csl.pl), Apr 05 2008
%Y A002997 Cf. A001567, A064238-A064262, A006931, A055553, A002322, A083737, A153581.
%Y A002997 Sequence in context: A047713 A006971 A104016 this_sequence A087788 A083733 
               A048123
%Y A002997 Adjacent sequences: A002994 A002995 A002996 this_sequence A002998 A002999 
               A003000
%K A002997 nonn,nice,easy
%O A002997 1,1
%A A002997 N. J. A. Sloane (njas(AT)research.att.com).
%E A002997 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

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