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| In [[number theory]], a '''Carmichael number''' is a [[composite number|composite]] positive [[integer]] <math>n</math> which satisfies the [[Modular arithmetic|congruence]]
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| :<math>b^{n}\equiv b\pmod{n}</math>
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| for all integers <math>1<b<n</math> . They are named for [[Robert Daniel Carmichael|Robert Carmichael]]. The Carmichael numbers are the [[Knödel number]]s ''K''<sub>1</sub>.
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| ==Overview==
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| [[Fermat's little theorem]] states that all [[prime numbers]] have the above property. In this sense, Carmichael numbers are similar to prime numbers; in fact, they are called [[Fermat pseudoprime]]s. Carmichael numbers are sometimes also called '''absolute Fermat pseudoprimes'''.
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| Carmichael numbers are important because they pass the [[Fermat primality test]] but are not actually prime. Since Carmichael numbers exist, this primality test cannot be relied upon to prove the primality of a number, although it can still be used to prove a number is composite. This makes tests based on Fermat's Little Theorem risky compared to other more stringent tests such as the [[Solovay-Strassen primality test]] or a [[strong pseudoprime]] test.
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| Still, as numbers become larger, Carmichael numbers become very rare. For example, there are 20,138,200 Carmichael numbers between 1 and 10<sup>21</sup> (approximately one in 50 trillion (50e12) numbers).<ref name="Pinch2007">Richard Pinch, [http://s369624816.websitehome.co.uk/rgep/p82.pdf "The Carmichael numbers up to 10<sup>21</sup>"], May 2007.</ref>
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| ===Korselt's criterion===
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| An alternative and equivalent definition of Carmichael numbers is given by '''Korselt's criterion'''.
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| :'''Theorem''' ([[Alwin Korselt|A. Korselt]] 1899): A positive composite integer <math>n</math> is a Carmichael number if and only if <math>n</math> is [[square-free integer|square-free]], and for all [[prime divisor]]s <math>p</math> of <math>n</math>, it is true that <math>p - 1 \mid n - 1</math>.
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| It follows from this theorem that all Carmichael numbers are [[odd number|odd]], since any even composite number that is square-free (and hence has only one prime factor of two) will have at least one odd prime factor, and thus <math>p - 1 \mid n - 1</math> results in an even dividing an odd, a contradiction. (The oddness of Carmichael numbers also follows from the fact that <math>-1</math> is a [[Fermat primality test|Fermat witness]] for any even composite number.)
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| From the criterion it also follows that Carmichael numbers are [[Cyclic number (group theory)|cyclic]].<ref>[http://www.numericana.com/data/crump.htm Carmichael Multiples of Odd Cyclic Numbers] "Any divisor of a Carmichael number must be an odd cyclic number"</ref><ref>Proof sketch: If <math>n</math> is square-free but not cyclic, <math>p_i \mid p_j - 1</math> for two prime factors <math>p_i</math> and <math>p_j</math> of <math>n</math>. But if <math>n</math> satisfies Korselt then <math>p_j - 1 \mid n - 1</math>, so by transitivity of the "divides" relation <math>p_i \mid n - 1</math>. But <math>p_i</math> is also a factor of <math>n</math>, a contradiction.</ref>
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| ==Discovery==
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| Korselt was the first who observed the basic properties of Carmichael numbers, but he could not find any examples. In 1910, Carmichael<ref name="Carmichael1910">{{cite journal |author=R. D. Carmichael|title=Note on a new number theory function |journal=Bulletin of the American Mathematical Society |volume=16 |issue=5|year=1910 |pages=232–238 |url=http://www.ams.org/journals/bull/1910-16-05/home.html}}</ref> found the first and smallest such number, 561, which explains the name "Carmichael number".
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| That 561 is a Carmichael number can be seen with Korselt's criterion. Indeed, <math>561 = 3 \cdot 11 \cdot 17</math> is square-free and <math>2 | 560</math>, <math>10 | 560</math> and <math>16 | 560</math>.
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| The next six Carmichael numbers are {{OEIS|id=A002997}}:
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| :<math>1105 = 5 \cdot 13 \cdot 17 \qquad (4 \mid 1104;\quad 12 \mid 1104;\quad 16 \mid 1104)</math>
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| :<math>1729 = 7 \cdot 13 \cdot 19 \qquad (6 \mid 1728;\quad 12 \mid 1728;\quad 18 \mid 1728)</math>
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| :<math>2465 = 5 \cdot 17 \cdot 29 \qquad (4 \mid 2464;\quad 16 \mid 2464;\quad 28 \mid 2464)</math>
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| :<math>2821 = 7 \cdot 13 \cdot 31 \qquad (6 \mid 2820;\quad 12 \mid 2820;\quad 30 \mid 2820)</math>
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| :<math>6601 = 7 \cdot 23 \cdot 41 \qquad (6 \mid 6600;\quad 22 \mid 6600;\quad 40 \mid 6600)</math>
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| :<math>8911 = 7 \cdot 19 \cdot 67 \qquad (6 \mid 8910;\quad 18 \mid 8910;\quad 66 \mid 8910).</math>
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| These first seven Carmichael numbers, from 561 to 8911, were all found by the Czech mathematician [[Václav Šimerka]] in 1885<ref name="Simerka1885">{{cite journal |author=V. Šimerka|title=Zbytky z arithmetické posloupnosti (On the remainders of an arithmetic progression) |journal=Časopis pro pěstování matematiky a fysiky |volume=14 |issue=5|year=1885 |pages=221–225 |url=http://dml.cz/handle/10338.dmlcz/122245}}</ref> (thus preceding not just Carmichael but also Korselt, although Šimerka did not find anything like Korselt's criterion). His work, however, remained unnoticed.
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| J. Chernick<ref name="Chernick1939">{{cite journal |author=Chernick, J. |title=On Fermat's simple theorem |journal=Bull. Amer. Math. Soc. |volume=45 |year=1939 |pages=269–274 |doi=10.1090/S0002-9904-1939-06953-X |url=http://www.ams.org/journals/bull/1939-45-04/S0002-9904-1939-06953-X/S0002-9904-1939-06953-X.pdf}}</ref> proved a theorem in 1939 which can be used to construct a [[subset]] of Carmichael numbers. The number <math>(6k + 1)(12k + 1)(18k + 1)</math> is a Carmichael number if its three factors are all prime. Whether this formula produces an infinite quantity of Carmichael numbers is an open question (though it is implied by [[Dickson's conjecture]]).
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| [[Paul Erdős]] heuristically argued there should be infinitely many Carmichael numbers. In 1994 it was shown by [[W. R. (Red) Alford]], [[Andrew Granville]] and [[Carl Pomerance]] that there really do exist infinitely many Carmichael numbers. Specifically, they showed that for sufficiently large <math>n</math>, there are at least <math>n^{2/7}</math> Carmichael numbers between 1 and <math>n</math>.<ref name="Alford1994">{{cite journal |author=[[W. R. (Red) Alford|W. R. Alford]] |coauthors=[[Andrew Granville]], [[Carl Pomerance]] |title=There are Infinitely Many Carmichael Numbers |journal=[[Annals of Mathematics]] |volume=139 |year=1994 |pages=703–722 |doi=10.2307/2118576 |url=http://www.math.dartmouth.edu/~carlp/PDF/paper95.pdf}}</ref>
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| Löh and Niebuhr in 1992 found some huge Carmichael numbers, including one with 1,101,518 factors and over 16 million digits.
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| ==Properties==
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| === Factorizations ===
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| Carmichael numbers have at least three positive prime factors. The first Carmichael numbers with <math>k = 3, 4, 5, \ldots</math> prime factors are {{OEIS|id=A006931}}:
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| {| class="wikitable"
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| |-
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| !''k'' !!
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| |-
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| | 3 || <math>561 = 3 \cdot 11 \cdot 17\,</math>
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| |-
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| | 4 || <math>41041 = 7 \cdot 11 \cdot 13 \cdot 41\,</math>
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| |-
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| | 5 || <math>825265 = 5 \cdot 7 \cdot 17 \cdot 19 \cdot 73\,</math>
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| |-
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| | 6 || <math>321197185 = 5 \cdot 19 \cdot 23 \cdot 29 \cdot 37 \cdot 137\,</math>
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| |-
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| | 7 || <math>5394826801 = 7 \cdot 13 \cdot 17 \cdot 23 \cdot 31 \cdot 67 \cdot 73\,</math>
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| |-
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| | 8 || <math>232250619601 = 7 \cdot 11 \cdot 13 \cdot 17 \cdot 31 \cdot 37 \cdot 73 \cdot 163\,</math>
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| |-
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| | 9 || <math>9746347772161 = 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 31 \cdot 37 \cdot 41 \cdot 641\,</math>
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| |}
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| The first Carmichael numbers with 4 prime factors are {{OEIS|id=A074379}}:
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| {| class="wikitable"
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| |-
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| !''i'' !!
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| |-
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| | 1 || <math>41041 = 7 \cdot 11 \cdot 13 \cdot 41\,</math>
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| |-
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| | 2 || <math>62745 = 3 \cdot 5 \cdot 47 \cdot 89\,</math>
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| |-
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| | 3 || <math>63973 = 7 \cdot 13 \cdot 19 \cdot 37\,</math>
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| |-
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| | 4 || <math>75361 = 11 \cdot 13 \cdot 17 \cdot 31\,</math>
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| |-
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| | 5 || <math>101101 = 7 \cdot 11 \cdot 13 \cdot 101\,</math>
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| |-
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| | 6 || <math>126217 = 7 \cdot 13 \cdot 19 \cdot 73\,</math>
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| |-
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| | 7 || <math>172081 = 7 \cdot 13 \cdot 31 \cdot 61\,</math>
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| |-
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| | 8 || <math>188461 = 7 \cdot 13 \cdot 19 \cdot 109\,</math>
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| |-
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| | 9 || <math>278545 = 5 \cdot 17 \cdot 29 \cdot 113\,</math>
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| |-
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| | 10 || <math>340561 = 13 \cdot 17 \cdot 23 \cdot 67\,</math>
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| |}
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| The second Carmichael number (1105) can be expressed as the sum of two squares in more ways than any smaller number. The third Carmichael number ([[1729 (number)|1729]]) is the Hardy-Ramanujan Number: the smallest number that can be expressed as the sum of two cubes in two different ways.
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| ===Distribution===
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| Let <math>C(X)</math> denote the number of Carmichael numbers less than or equal to <math>X</math>. The distribution of Carmichael numbers by powers of 10:<ref name="Pinch2007"/>
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| <center>
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| {| class="wikitable"
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| |-
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| ! <math>n</math>
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| | 3
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| | 4
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| | 5
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| | 6
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| | 7
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| | 8
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| | 9
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| | 10
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| | 11
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| | 12
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| | 13
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| | 14
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| | 15
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| | 16
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| | 17
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| | 18
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| | 19
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| | 20
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| | 21
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| |-
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| ! <math>C(10^n)</math>
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| | 1
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| | 7
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| | 16
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| | 43
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| | 105
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| | 255
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| | 646
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| | 1547
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| | 3605
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| | 8241
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| | 19279
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| | 44706
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| | 105212
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| | 246683
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| | 585355
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| | 1401644
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| | 3381806
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| | 8220777
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| | 20138200
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| |}
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| </center>
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| In 1953, Knödel proved the [[Upper and lower bounds|upper bound]]:
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| :<math>C(X) < X \exp\left({-k_1 \left( \log X \log \log X\right)^\frac{1}{2}}\right)</math>
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| for some constant <math>k_1</math>.
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| In 1956, Erdős improved the bound to<ref name="Erdős1956">{{cite journal |author=[[Paul Erdős|Erdős, P.]] |year=1956 |title=On pseudoprimes and Carmichael numbers |journal=Publ. Math. Debrecen |volume=4 |pages=201–206 |url=http://www.renyi.hu/~p_erdos/1956-10.pdf |mr=79031 }}</ref>
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| :<math>C(X) < X \exp\left(\frac{-k_2 \log X \log \log \log X}{\log \log X}\right)</math>
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| for some constant <math>k_2</math>. He further gave a [[heuristic argument]] suggesting that this upper bound should be close to the true growth rate of <math>C(X)</math>. The table below gives approximate minimal values for the constant ''k'' in the Erdős bound for <math>X=10^n</math> as ''n'' grows:
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| <center> | |
| {| class="wikitable"
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| |-
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| ! <math>n</math>
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| | 4
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| | 6
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| | 8
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| | 10
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| | 12
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| | 14
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| | 16
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| | 18
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| | 20
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| | 21
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| |-
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| ! ''k''
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| | 2.19547
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| | 1.97946
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| | 1.90495
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| | 1.86870
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| | 1.86377
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| | 1.86293
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| | 1.86406
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| | 1.86522
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| | 1.86598
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| | 1.86619
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| |}
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| </center>
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| In the other direction, [[W. R. (Red) Alford|Alford]], [[Andrew Granville|Granville]] and [[Carl Pomerance|Pomerance]] proved in 1994<ref name="Alford1994"/> that for sufficiently large ''X'',
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| :<math>C(X) > X^{2/7}.</math>
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| In 2005, this bound was further improved by [[Glyn Harman|Harman]]<ref>{{cite journal |author=Glyn Harman |title=On the number of Carmichael numbers up to ''x'' |journal=Bulletin of the London Mathematical Society |volume=37 |year=2005 |pages=641–650 |doi=10.1112/S0024609305004686}}</ref> to
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| :<math>C(X) > X^{0.332}</math>
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| and then has subsequently improved the exponent to just over <math>1/3</math>. | |
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| Regarding the asymptotic distribution of Carmichael numbers, there have been several conjectures. In 1956, Erdős<ref name="Erdős1956"/> conjectured that there were <math>X^{1-o(1)}</math> Carmichael numbers for ''X'' sufficiently large. In 1981, Pomerance<ref name="Pomerance1981">{{cite journal |author=[[Carl Pomerance|Pomerance, C.]] |year=1981 |title=On the distribution of pseudoprimes |journal=Math. Comp. |volume=37 |pages=587–593|jstor=2007448}}</ref> sharpened Erdős' heuristic arguments to conjecture that there are
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| :<math>X^{1-{\frac{\{1+o(1)\}\log\log\log X}{\log\log X}}}</math>
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| Carmichael numbers up to ''X''. However, inside current computational ranges (such as the counts of Carmichael numbers performed by Pinch<ref name="Pinch2007"/> up to 10<sup>21</sup>), these conjectures are not yet borne out by the data.
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| ==Generalizations==
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| The notion of Carmichael number generalizes to a Carmichael ideal in any number field ''K''. For any nonzero prime ideal <math>\mathfrak p</math> in <math>{\mathcal O}_K</math>, we have <math>\alpha^{{\rm N}(\mathfrak p)} \equiv \alpha \bmod {\mathfrak p}</math> for all <math>\alpha</math> in <math>{\mathcal O}_K</math>, where <math>{\rm N}(\mathfrak p)</math> is the norm of the ideal <math>\mathfrak p</math>. (This generalizes Fermat's little theorem, that <math>m^p \equiv m \bmod p</math> for all integers ''m'' when ''p'' is prime.) Call a nonzero ideal <math>\mathfrak a</math> in <math>{\mathcal O}_K</math> Carmichael if it is not a prime ideal and <math>\alpha^{{\rm N}(\mathfrak a)} \equiv \alpha \bmod {\mathfrak a}</math> for all <math>\alpha \in {\mathcal O}_K</math>, where <math>{\rm N}(\mathfrak a)</math> is the norm of the ideal <math>\mathfrak a</math>. When ''K'' is <math>\mathbf Q</math>, the ideal <math>\mathfrak a</math> is principal, and if we let ''a'' be its positive generator then the ideal <math>\mathfrak a = (a)</math> is Carmichael exactly when ''a'' is a Carmichael number in the usual sense.
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| When ''K'' is larger than the rationals it is easy to write down Carmichael ideals in <math>{\mathcal O}_K</math>: for any prime number ''p'' that splits completely in ''K'', the principal ideal <math>p{\mathcal O}_K</math> is a Carmichael ideal. Since infinitely many prime numbers split completely in any number field, there are infinitely many Carmichael ideals in <math>{\mathcal O}_K</math>. For example, if ''p'' is any prime number that is 1 mod 4, the ideal (''p'') in the Gaussian integers '''Z'''[''i''] is a Carmichael ideal.
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| Both prime and Carmichael numbers satisfy the following equality:
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| :<math>\gcd \left(\sum_{x=1}^{n-1} x^{n-1}, n\right)=1</math>
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| ==Higher-order Carmichael numbers==
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| Carmichael numbers can be generalized using concepts of [[abstract algebra]].
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| The above definition states that a composite integer ''n'' is Carmichael
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| precisely when the ''n''th-power-raising function ''p''<sub>''n''</sub> from the [[ring (mathematics)|ring]] '''Z'''<sub>''n''</sub> of integers modulo ''n'' to itself is the identity function. The identity is the only '''Z'''<sub>''n''</sub>-[[algebra over a field|algebra]] [[endomorphism]] on '''Z'''<sub>''n''</sub> so we can restate the definition as asking that ''p''<sub>''n''</sub> be an algebra endomorphism of '''Z'''<sub>''n''</sub>.
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| As above, ''p''<sub>''n''</sub> satisfies the same property whenever ''n'' is prime.
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| The ''n''th-power-raising function ''p''<sub>''n''</sub> is also defined on any '''Z'''<sub>''n''</sub>-algebra '''A'''. A theorem states that ''n'' is prime if and only if all such functions ''p''<sub>''n''</sub> are algebra endomorphisms.
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| In-between these two conditions lies the definition of '''Carmichael number of order m''' for any positive integer ''m'' as any composite number ''n'' such that ''p''<sub>''n''</sub> is an endomorphism on every '''Z'''<sub>''n''</sub>-algebra that can be generated as '''Z'''<sub>''n''</sub>-[[module (mathematics)|module]] by ''m'' elements. Carmichael numbers of order 1 are just the ordinary Carmichael numbers.
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| ===Properties===
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| Korselt's criterion can be generalized to higher-order Carmichael numbers, as shown by Howe.<ref>Everett W. Howe. [http://arxiv.org/abs/math.NT/9812089 "Higher-order Carmichael numbers."] ''Mathematics of Computation'' '''69''' (2000), pp. 1711–1719.</ref>
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| A heuristic argument, given in the same paper, appears to suggest that there are infinitely many Carmichael numbers of order ''m'', for any ''m''. However, not a single Carmichael number of order 3 or above is known.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite journal |author=Carmichael, R. D.|year=1910|title=Note on a new number theory function |journal=[[Bulletin of the American Mathematical Society]] |volume=16 |issue=5|pages=232–238 |url=http://www.ams.org/journals/bull/1910-16-05/home.html}}
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| *{{cite journal |author=Carmichael, R. D. |year=1912 |title=On composite numbers ''P'' which satisfy the Fermat congruence <math>a^{P-1}\equiv 1\bmod P</math> |journal=[[American Mathematical Monthly]] |volume=19 |issue=2 |pages=22–27 |doi=10.2307/2972687}}
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| *{{cite journal |author=Chernick, J. |year=1939 |title=On Fermat's simple theorem |journal=Bull. Amer. Math. Soc. |volume=45 |pages=269–274 |doi=10.1090/S0002-9904-1939-06953-X |url=http://www.ams.org/journals/bull/1939-45-04/S0002-9904-1939-06953-X/S0002-9904-1939-06953-X.pdf}}
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| *{{cite journal |author=Korselt, A. R. |year=1899 |title=Problème chinois |journal=[[L'Intermédiaire des Mathématiciens]] |volume=6 |pages=142–143}}
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| *{{cite journal |author=Löh, G.; Niebuhr, W. |year=1996 |url=http://www.ams.org/mcom/1996-65-214/S0025-5718-96-00692-8/S0025-5718-96-00692-8.pdf |title=A new algorithm for constructing large Carmichael numbers |journal=Math. Comp. |volume=65 |pages=823–836 |doi=10.1090/S0025-5718-96-00692-8}}
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| *{{cite book | title = The Book of Prime Number Records | publisher = Springer | year = 1989 | isbn = 978-0-387-97042-4 | author = [[Paulo Ribenboim|Ribenboim, P.]] }}
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| *{{cite journal |author=Šimerka, V.|year=1885 |title=Zbytky z arithmetické posloupnosti (On the remainders of an arithmetic progression) |journal=Časopis pro pěstování matematiky a fysiky |volume=14 |issue=5 |pages=221–225 |url=http://dml.cz/handle/10338.dmlcz/122245}}
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| ==External links==
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| *{{springer|title=Carmichael number|id=p/c110100}}
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| *[http://de.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Carmichael-Zahlen Table of Carmichael numbers]
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| *[http://www.kobepharma-u.ac.jp/~math/notes/note02.html Carmichael numbers up to 10^12]
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| *{{MathPages|id=home/kmath028/kmath028|title=The Dullness of 1729}}
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| *{{MathWorld | urlname=CarmichaelNumber | title=Carmichael Number}}
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| *[http://www.numericana.com/answer/modular.htm Final Answers Modular Arithmetic]
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| {{Classes of natural numbers}}
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| [[Category:Integer sequences]]
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| [[Category:Modular arithmetic]]
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| [[Category:Pseudoprimes]]
| |
The quality of grass and soil in your paddock is of particular importance both to alter the your land and the heath of the horses. As a result it is much better to check the conditions of your paddock frequently and treat any problems as they arise. Developing is to write aspect making and maintaining healthy grassland is spin. Often horses graze and manure in specific associated with the field, leaving bare patches of land. Most certainly paddocks end up being left empty for 6 months as upkeep is necessary. However, in practice this is not always feasible. As such it become more practical to section off one small area of land at a period for longer. This is especially important through the autumn and summer the seasons.
Start buying your materials in portions. Few people can afford in order to purchase everything with their projects at one time. By dividing projects into multiple phases, within your budget what basic ingredients as going. This may save you money, assist you keep track of your progress, and enable you to adjust plans prior at your next consideration.
When choosing a home, together with every monthly payment, in order to putting money into final savings. On the contrary, when you are renting a home you for you to give money to the owner it a great expense. Any time you pay a mortgage, a percentage goes toward your value guarantee. Indeed, this is similar to having money inside your bank account, which you can draw upon later anytime. Every year, rental rates increase as well as the principle on your mortgage is decreasing collectively payment. Further, the housing sector continues to grow and hence the valuations of family homes.
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Keep your trees and hedges trimmed, keep your lawn well manicured, remember the grass well watered, edge the areas between the sidewalk as well as the grass - all the work will improve a good first impression of residence. If you've got a dog, make certain that the lawn is clean. Weed the flower garden, whilst your deck or patio clean and clear of debris.
There will be no better or cheaper way to establish a good first impression when compared with buying a new, clean doormat. First impressions are absolutely crucial, and an early worn out doormat is really a telltale manifestation of owner leave out. Also, a good doormat will stop mud from being tracked into your own home during the show intervals.
Estimate the cost of your project before inexperienced. First, write down a list in the needed materials and goods. Then, consider where best to purchase all individuals items. Child overspend, which is can happen easily since different stores can have huge variations in price depending on where however located. Discover where to get top material at the minimum price.
There is just not substitute nutrition landscape design when it will come to making a beautiful abode. Because there is so much information available around the topic, can make sense to invest the time necessary to find the ideas and concepts most relevant to your own specific requests. The guidance in this particular piece can be a perfect place to get started.
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