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Topic: Carmichael number

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Odd number

Prime number

Fermat primality test

Euler pseudoprime

Fibonacci pseudoprime

Modular arithmetic theory

Robert Daniel Carmichael

Modular arithmetic

Modular exponentiation

Coprime

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	Encyclopedia: Prime number (Site not responding. Last check: 2007-10-21)
		is the number of planar partitions of 10.
		is the number of planar partitions of 11.
		is the number of planar partitions of 12.
	www.nationmaster.com /encyclopedia/Prime-number (1027 words)

	What's Special About This Number?
		is the number of planar partitions of 13.
		is the number of planar partitions of 14.
		is the number of planar partitions of 16.
	www.stetson.edu /~efriedma/numbers.html (7277 words)

	Modular Arithmetic, Fermat Theorem, Carmichael Numbers - Numericana
		A Carmichael number is a composite number n dividing a
		Thus, 1729 is a Carmichael number because its prime factorization is 7.13.19 while 1728 happens to be divisible by 6, 12 and 18.
		Generic Carmichael numbers are obtained from the methods presented in the preceding article when several predetermined numbers are prime, a condition which is rarely satisfied when such numbers are extremely large.
	home.att.net /~numericana/answer/modular.htm (3113 words)

	Zeisel number - Wikipedia, the free encyclopedia
		A Zeisel number is a square-free integer k with at least three prime factors which fall into the pattern
		where a and b are some integer constants and x is the index number of each prime factor in the factorization, sorted from lowest to highest.
		Other Carmichael numbers of that kind are: 294409, 56052361, 118901521, 172947529, 216821881, 228842209, 1299963601, 2301745249, 9624742921,...
	en.wikipedia.org /wiki/Zeisel_number (273 words)

	Notable Properties of Specific Numbers at MROB
		Let anyone who has intelligence (penetration and insight enough) calculate the number of the beast, for it is a human number [the number of a certain man]; his number is six-hundred sixty six.
		There are two ways to interpret the contradiction: either the number itself is unspecified (and 666 is just a metaphor), or else 666 is indeed the answer but that the method of calculation to produce it (and/or perhaps the identity of the "certain man") is unspecfied.
		Numbers with lots of factors were popular in ancient civilizations; well-known examples include 12, 24, 60 and 360.
	home.earthlink.net /~mrob/pub/math/numbers-8.html (3200 words)

	Carmichael number - InfoSearchPoint.com (Site not responding. Last check: 2007-10-21)
		Carmichael numbers are important because they can fool the Fermat primality test.
		Theorem (Korselt): A positive and odd integer N is a Carmichael number if and only if N is square-free, and for all prime divisors p of N, p-1 divides N-1.
		It follows from this theorem that Carmichael numbers are always odd.
	www.infosearchpoint.com /display/Carmichael_number (750 words)

	Number Theory Glossary
		A Carmichael Number is a composite number which passes the Fermat pseudoprime test for all bases.
		A number which is not composite is called prime.
		A prime number is a number which has no factors other than 1 (called non-trivial factors).
	www.math.umbc.edu /~campbell/NumbThy/Class/Glossary.html (827 words)

	Carmichael Number
		Carmichael number is composite number that passed probabilistic prime tests.
		Carmichael number will pass Fermat test, Lucas test, Miller-Rabin test and any other probabilistic prime test.
		NOTE: Normally to avoid Carmichael number, developer use many base values (witnesses), but developer use fewer base values (witnesses) to keep the speed of the running test so it will not hurt the performance.
	www.geocities.com /hmaxf_urlcr/carmichael.htm (182 words)

	Pseudoprimes and Carmichael numbers
		The numbers were generated by a variety of strategies, the most important being a back-tracking search for possible prime factorisations, and the computations checked by a sieving technique..
		The numbers were generated by a back-tracking search for possible prime factorisations, and the computations checked by searching selected ranges of integers directly using a sieving technique, together with a ``large prime variation''.
		We show that the inadvertent use of a Carmichael number instead of a prime factor in the modulus of an RSA cryptosystem is likely to make the system fatally vulnerable, but that such numbers may be detected.
	www.chalcedon.demon.co.uk /rgep/carpsp.html (448 words)

	Carmichael numbers with a large number of prime factors (Site not responding. Last check: 2007-10-21)
		Carmichael numbers with a large number of prime factors
		A Carmichael number is known to be the product of k distinct odd prime factors p_1, p_2,...
		Carmichaels with even more factors are being found on a SIEMENS 7.882 mainframe.
	www.rrz.uni-hamburg.de /loeh/abstr/ab1988.html (160 words)

	Primality testing
		The objective of primality testing is to take as input a positive integer, and return as output a verdict whether the number is prime or composite.
		In case ``Composite'' is returned, we know that the number is composite, because a proof of compositeness is produced.
		Carmichael numbers are rare, but it has been proved recently that there are infinitely many of them (Alford, Granville, Pomerance, 1994).
	people.cs.uchicago.edu /~laci/reu03/notes11/node3.html (519 words)

	Carmichael Numbers (Site not responding. Last check: 2007-10-21)
		Let a be a random number between 2 and n - 1 (being n the number whose primality we are testing).
		Some numbers that are not prime still pass the Fermat test with every number smaller than themselves.
		For each number in the input, you have to print if it is a Carmichael number or not, as shown in the sample output.
	acm.uva.es /p/v100/10006.html (323 words)

	math lessons - Carmichael number (Site not responding. Last check: 2007-10-21)
		In number theory, a Carmichael number is a composite positive integer n which satisfies the congruence b
		The first Carmichael numbers with k = 3, 4, 5, … prime factors are (sequence A006931 in OEIS):
		The first Carmichael numbers with 4 prime factors are (sequence A074379 in OEIS):
	www.mathdaily.com /lessons/Carmichael_number (878 words)

	Numbers
		Two numbers n and m are called an amicable pair if the sum of all positive divisors of n is equal to the sum of all positive divisors of m and both are equal to n + m.
		Carmichael numbers behave like prime numbers with respect to the most useful primality test, that is they pretend to be prime.
		Pentagonal numbers to pentagons is the same as triangular numbers to triangles and square numbers to squares.
	www.tanyakhovanova.com /Numbers/numbers.html (1912 words)

	Puzzle 171. Perfect & Carmichael numbers
		The value a_t+43216624217 is the minimal titanic Carmichael for the perfect number 28.
		This number was found with a custom written 5-way modular sieve, and 5-prp test built in.
		The numbers aren't proved prime, but have passed 2 different fermat tests and 10 MR tests each.
	www.primepuzzles.net /puzzles/puzz_171.htm (628 words)

	Number theory algorithms (Site not responding. Last check: 2007-10-21)
		Checking Carmichael numbers is slow, but it is easy to show that if n is a large enough prime number, then neither 3n+1, nor 8n+1, nor any s*n+1 with small integer s can be a perfect square.
		The number of levels of recursion is logarithmic in the arguments a, b.
		A "Gaussian integer" is a complex number of the form z=a+b*I, where a and b are ordinary (rational) integers.
	yacas.sourceforge.net /Algochapter2.html (3917 words)

	Higher-order Carmichael numbers, by Everett W. Howe (Site not responding. Last check: 2007-10-21)
		We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZ-algebra that can be generated as a Z/nZ-module by m elements.
		We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdos for the usual Carmichael numbers) that indicates that for every m there should be infinitely many Carmichael numbers of order m.
		The argument suggests a method for finding examples of higher-order Carmichael numbers; we use the method to provide examples of Carmichael numbers of order 2.
	www.math.uiuc.edu /Algebraic-Number-Theory/0160 (119 words)

	Math Forum - Ask Dr. Math Archives: College Number Theory
		Apparently almost any number can be made into a palindromic number by reversing the digits and adding and then repeating the steps until you get a palindromic number.
		An antifirst number is a number with more divisors than every number before it.
		Is it possible to find a formula that takes X (a large number) and creates another expression which equals X but is shorter in length than X? For example, take 390,625 which uses six characters.
	mathforum.org /library/drmath/sets/college_number_theory.html (995 words)

	Problem 2: Fermat Primality Testing (Site not responding. Last check: 2007-10-21)
		To test that a number, p, is prime, we compute a^(p - 1) (mod p) where a is a number between 2 and p - 1, inclusive and a is coprime to p.
		Unfortunately, the test is not air tight as there are numbers, known as Carmichael numbers, which pass the test for every possible value of a yet are not prime.
		If p fails the test for any number then print p followed by "is not a prime number".
	www.eecis.udel.edu /~breech/contest.inet.fall.03/problems/fermat.html (236 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		For all nonnegative integers m & n all numbers of the form f(m,n) = (100(610^m - 1)+ 40)(10^((m + 2)n) - 1)/(10^(m + 2) - 1) are in the sequence, in fact f(m,n) = (5.(9)(m))(n).0 where dot between numbers means concatenation and "(r)(t)" means number of r's is t.
		While Carmichael numbers seem to be Devaraj numbers (though this is not proved [added later by A. Devaraj, dkandadai(AT)yahoo.com: this may now have been proved]), the opposite is not true.
		A Carmichael number is an an odd composite number n which is a pseudoprime to base a for every number a prime to n.
	www.research.att.com:9000 /~njas/sequences/eisBTfry00108.txt (6174 words)

	Conjecture 19. A bound to the largest prime factor of certain Carmichael numbers
		"For Carmichael numbers with exactly three prime factors, once one of the primes has been specified, there are only a finite number of Carmichael numbers which can be constructed.
		So there are only a fixed number of possible values of D. For each value of D, there are only finitely many values of a and b, since ab = D+p^2.
		And indeed, 3 1117 is a Carmichael number, so this bound is attained.
	www.primepuzzles.net /conjectures/conj_019.htm (1131 words)

	Number theory (Site not responding. Last check: 2007-10-21)
		Also, Carmichael numbers must be odd and have at least three prime factors.
		Although these numbers are rare (there are only 43 such numbers between 1 and 10^6), it has recently been proven that there are infinitely many of them.
		A pair of numbers m, n has this property if the sum of the proper divisors of m is n and the sum of the proper divisors of n is m.
	yacas.sourceforge.net /refchapter4.html (1987 words)

	The Carmichael Numbers Up To (ResearchIndex)
		As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a "large prime variation".
		We present further statistics on the distribution of Carmichael numbers.
		Introduction A Carmichael number N is a composite number N with the property that for every b prime to N we have b N \Gamma1 j 1 mod N.
	citeseer.ist.psu.edu /391217.html (236 words)

	[No title]
		The expalnation of the remarkable behaviour of the number n=225593397919 is that the above condition holds for every coprime base b.
		Odd composite numbers n with the property that they are pseudoprime to any coprime base are called Carmichael numbers.
		The converse is also true: every Carmichael number n has at least three prime factors, all its prime factors p are different and satisfy
	www.ma.umist.ac.uk /avb/117ws9.html (610 words)

	math lessons - Euler-Jacobi pseudoprime (Site not responding. Last check: 2007-10-21)
		In number theory, an odd composite integer n is called an Euler-Jacobi pseudoprime to base a, if a and n are coprime, and
		The motivation for this definition is the fact that all prime numbers n satisfy the above equation, as explained in the Legendre symbol article.
		There are no numbers which are Euler-Jacobi pseudoprimes to all bases as Carmichael numbers are.
	www.mathdaily.com /lessons/Euler-Jacobi_pseudoprime (433 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Thus, for example, not only is (6x+1)(12x+1)(18x+1) a Carmichael number whenever the three factors are prime, but also (6x+1)(12x+1)(18x+1)(36x+1) is a Carmichael number if in addition 36x+1 is prime.
		There are numbers N for which there are two (or more) distinct sets of divisors whose sums are divisible by N. (By distinct, I mean having no elements in common.) For example, 120 = 20+40+60 = 1+2+3+4+5+6+8+10+12+15+24+30.
		If we can find an x such that the corresponding p[i] are all prime, then we will have found a Carmichael number which is the product of two smaller Carmichael numbers.
	www.ics.uci.edu /~eppstein/numth/egypt/carmichael.html (377 words)

	MAT 312/AMS 351 - Applied Algebra -- Spring 2005
		Show Lam\'e's result that the number of divisions needed to find the gcd of two integers via the euclidean algorithm does not exceed the number of decimal digits in the smaller of the two integers.
		Use this to show that 564651361 is a Carmichael number.
		Hardy and Littlewood conjectured that the number of twin primes not exceeding
	www.math.sunysb.edu /~sorin/312/projects.shtml (856 words)

	Carmichael numbers
		If a number is a pseudoprime to a variety of bases, then it is likely to be a prime.
		A Carmichael number is a composite number n such that b
		So a Carmichael number passes the Fermat's-little-theorem test as best as it can.
	www.math.fau.edu /Richman/carm.htm (110 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Subject: Carmichael Numbers: Korselt's Criterion [was: BuddhaSon number & Carmichael number] Date: 26 Mar 1999 08:14:31 -0500 Newsgroups: sci.math fredh@ix.netcom.com (Fred W. Helenius) wrote:
		It was not until 11 years later that Carmichael actually gave some examples while presenting his essentially equivalent criterion that y(n)n-1 (and n is composite), where y(n) is the Carmichael lambda function, the (universal) exponent of the group of units in Z/n, i.e.
		QED For more on Carmichael numbers see the fine expository survey by Carl Pomerance: Carmichael Numbers, Nieuw Archief voor Wiskunde, 11 (1993) 199-209 (an elaboration of his April 22, 1992 Beeger Lecture during the 28th Nederlands Math.
	www.math.niu.edu /~rusin/known-math/99/korseldt (400 words)

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