Editing Carmichael number (section)

== Discovery ==
The first seven Carmichael numbers, from 561 to 8911, were all found by the Czech mathematician [[Václav Šimerka]] in 1885<ref name="Simerka1885">{{cite journal |last1=Šimerka|first1=Václav|author-link1=Václav Šimerka|title=Zbytky z arithmetické posloupnosti|trans-title=On the remainders of an arithmetic progression|journal=Časopis pro pěstování mathematiky a fysiky |volume=14 |issue=5|year=1885 |pages=221–225 |doi=10.21136/CPMF.1885.122245 |url=http://dml.cz/handle/10338.dmlcz/122245|doi-access=free }}</ref> (thus preceding not just Carmichael but also Korselt, although Šimerka did not find anything like Korselt's criterion).<ref>{{cite journal |last1=Lemmermeyer |first1=F. |title=Václav Šimerka: quadratic forms and factorization |journal=LMS Journal of Computation and Mathematics |date=2013 |volume=16 |pages=118–129 |doi=10.1112/S1461157013000065 |doi-access=free}}</ref> His work, published in Czech scientific journal ''[[Časopis pro pěstování matematiky a fysiky]]'', however, remained unnoticed.[[File:Vaclav Simerka.jpg|thumb|Václav Šimerka listed the first seven Carmichael numbers]]

Korselt was the first who observed the basic properties of Carmichael numbers, but he did not give any examples.

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 {{tmath|1= 2 \mid 560 }}, <math>10 \mid 560</math> and {{tmath|1= 16 \mid 560 }}.
The next six Carmichael numbers are {{OEIS|id=A002997}}:
: <math>1105 = 5 \cdot 13 \cdot 17 \qquad (4 \mid 1104;\quad 12 \mid 1104;\quad 16 \mid 1104)</math>
: <math>1729 = 7 \cdot 13 \cdot 19 \qquad (6 \mid 1728;\quad 12 \mid 1728;\quad 18 \mid 1728)</math>
: <math>2465 = 5 \cdot 17 \cdot 29 \qquad (4 \mid 2464;\quad 16 \mid 2464;\quad 28 \mid 2464)</math>
: <math>2821 = 7 \cdot 13 \cdot 31 \qquad (6 \mid 2820;\quad 12 \mid 2820;\quad 30 \mid 2820)</math>
: <math>6601 = 7 \cdot 23 \cdot 41 \qquad (6 \mid 6600;\quad 22 \mid 6600;\quad 40 \mid 6600)</math>
: <math>8911 = 7 \cdot 19 \cdot 67 \qquad (6 \mid 8910;\quad 18 \mid 8910;\quad 66 \mid 8910).</math>

In 1910, Carmichael himself<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=https://www.ams.org/journals/bull/1910-16-05/home.html |doi=10.1090/s0002-9904-1910-01892-9|doi-access=free }}</ref> also published the smallest such number, 561, and the numbers were later named after him.

[[Jack Chernick]]<ref name="Chernick1939">{{cite journal |author=Chernick, J. |title=On Fermat's simple theorem |journal=Bull. Amer. Math. Soc. |volume=45 |issue=4 |year=1939 |pages=269–274 |doi=10.1090/S0002-9904-1939-06953-X  |url=https://www.ams.org/journals/bull/1939-45-04/S0002-9904-1939-06953-X/S0002-9904-1939-06953-X.pdf|doi-access=free }}</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]]).

[[Paul Erdős]] heuristically argued there should be infinitely many Carmichael numbers. In 1994 [[W. R. (Red) Alford]], [[Andrew Granville]] and [[Carl Pomerance]]  used a bound on [[Olson's constant]] to show 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 {{tmath|1= n }}.<ref name="Alford-1994" />

[[Thomas Wright (mathematician)|Thomas Wright]] proved that if <math>a</math> and <math>m</math> are relatively prime,
then there are infinitely many Carmichael numbers in the [[arithmetic progression]] {{tmath|1= a + k \cdot m }},
where {{tmath|1= k = 1, 2, \ldots }}.<ref>{{cite journal |author=Thomas Wright |title=Infinitely many Carmichael Numbers in Arithmetic Progressions |journal=[[Bull. London Math. Soc.]] |volume=45 |year=2013 |issue=5 |pages=943–952 |arxiv=1212.5850 |doi=10.1112/blms/bdt013|s2cid=119126065 }}</ref>

Löh and Niebuhr in 1992 found some very large Carmichael numbers, including one with 1,101,518 factors and over 16 million digits.
This has been improved to 10,333,229,505 prime factors and 295,486,761,787 digits,<ref>{{cite journal |author1=W.R. Alford|author2=Jon Grantham|author3=Steven Hayman|author4=Andrew Shallue |display-authors=1|title=Constructing Carmichael numbers through improved subset-product algorithms|journal=Math. Comp.|volume=83|year=2014|issue=286|pages=899–915|arxiv=1203.6664|doi=10.1090/S0025-5718-2013-02737-8|s2cid=35535110|author-link=W. R. (Red) Alford}}</ref> so the largest known Carmichael number is much greater than the [[largest known prime]].