Carmichael number

Fermat's little theorem states that if $p$ is a prime number, then for any integer {{tmath|1= b }}, the number $b^p-b$ is an integer multiple of {{tmath|1= p }}. Carmichael numbers are composite numbers which have the same property. Carmichael numbers are also called Fermat pseudoprimes or absolute Fermat pseudoprimes. A Carmichael number will pass a Fermat primality test to every base $b$ relatively prime to the number, even though it is not actually prime.

This makes tests based on Fermat's Little Theorem less effective than strong probable prime tests such as the Baillie–PSW primality test and the Miller–Rabin primality test.

However, no Carmichael number is either an Euler–Jacobi pseudoprime or a strong pseudoprime to every base relatively prime to it

{{cite journal |author=D. H. Lehmer |author-link=Derrick Henry Lehmer |title=Strong Carmichael numbers |journal=J. Austral. Math. Soc. |date=1976 |volume=21 |issue=4 |pages=508–510 |doi=10.1017/s1446788700019364|doi-access=free }} Lehmer proved that no Carmichael number is an Euler-Jacobi pseudoprime to every base relatively prime to it. He used the term strong pseudoprime, but the terminology has changed since then. Strong pseudoprimes are a subset of Euler-Jacobi pseudoprimes. Therefore, no Carmichael number is a strong pseudoprime to every base relatively prime to it.

so, in theory, either an Euler or a strong probable prime test could prove that a Carmichael number is, in fact, composite.

Arnault

{{cite journal|title=Constructing Carmichael Numbers Which Are Strong Pseudoprimes to Several Bases|journal=Journal of Symbolic Computation|date=August 1995|volume=20|issue=2|pages=151–161

|author=F. Arnault|doi=10.1006/jsco.1995.1042|doi-access=free}}

gives a 397-digit Carmichael number $N$ that is a strong pseudoprime to all prime bases less than 307:

: $N\; =\; p\; \backslash cdot\; (313(p\; -\; 1)\; +\; 1)\; \backslash cdot\; (353(p\; -\; 1)\; +\; 1\; )$

where

: $p\; =${{hsp}}2{{hsp}}9674495668{{hsp}}6855105501{{hsp}}5417464290{{hsp}}5332730771{{hsp}}9917998530{{hsp}}4335099507{{hsp}}5531276838{{hsp}}7531717701{{hsp}}9959423859{{hsp}}6428121188{{hsp}}0336647542{{hsp}}1834556249{{hsp}}3168782883

is a 131-digit prime. $p$ is the smallest prime factor of {{tmath|1= N }}, so this Carmichael number is also a (not necessarily strong) pseudoprime to all bases less than {{tmath|1= p }}.

As numbers become larger, Carmichael numbers become increasingly rare. For example, there are 20,138,200 Carmichael numbers between 1 and 10²¹ (approximately one in 50 trillion (5·10¹³) numbers).

{{cite conference |url=http://tucs.fi/publications/attachment.php?fname=G46.pdf

|title=The Carmichael numbers up to 10²¹ |last=Pinch |first=Richard |date=December 2007

|editor=Anne-Maria Ernvall-Hytönen |volume=46

|publisher=Turku Centre for Computer Science |pages=129–131 |location=Turku, Finland

|conference=Proceedings of Conference on Algorithmic Number Theory

|access-date=2017-06-26

}}

An alternative and equivalent definition of Carmichael numbers is given by Korselt's criterion.

: Theorem (A. Korselt 1899): A positive composite integer $n$ is a Carmichael number if and only if $n$ is square-free, and for all prime divisors $p$ of {{tmath|1= n }}, it is true that {{tmath|1= p - 1 \mid n - 1 }}.

It follows from this theorem that all Carmichael numbers are 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 $p-1\; \backslash mid\; n-1$ results in an even dividing an odd, a contradiction. (The oddness of Carmichael numbers also follows from the fact that $-1$ is a Fermat witness for any even composite number.)

From the criterion it also follows that Carmichael numbers are cyclic.[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"Proof sketch: If $n$ is square-free but not cyclic, $p\_i\; \backslash mid\; p\_j\; -\; 1$ for two prime factors $p\_i$ and $p\_j$ of $n$. But if $n$ satisfies Korselt then {{tmath|1= p_j - 1 \mid n - 1 }}, so by transitivity of the "divides" relation {{tmath|1= p_i \mid n - 1 }}. But $p\_i$ is also a factor of {{tmath|1= n }}, a contradiction. Additionally, it follows that there are no Carmichael numbers with exactly two prime divisors.

The first seven Carmichael numbers, from 561 to 8911, were all found by the Czech mathematician Václav Šimerka in 1885{{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 }} (thus preceding not just Carmichael but also Korselt, although Šimerka did not find anything like Korselt's criterion).{{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}} His work, published in Czech scientific journal Časopis pro pěstování matematiky a fysiky, however, remained unnoticed.File:Vaclav Simerka.jpg

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, $561\; =\; 3\; \backslash cdot\; 11\; \backslash cdot\; 17$ is square-free and {{tmath|1= 2 \mid 560 }}, $10\; \backslash mid\; 560$ and {{tmath|1= 16 \mid 560 }}.

The next six Carmichael numbers are {{OEIS|id=A002997}}:

: $1105\; =\; 5\; \backslash cdot\; 13\; \backslash cdot\; 17\; \backslash qquad\; (4\; \backslash mid\; 1104;\backslash quad\; 12\; \backslash mid\; 1104;\backslash quad\; 16\; \backslash mid\; 1104)$

: $1729\; =\; 7\; \backslash cdot\; 13\; \backslash cdot\; 19\; \backslash qquad\; (6\; \backslash mid\; 1728;\backslash quad\; 12\; \backslash mid\; 1728;\backslash quad\; 18\; \backslash mid\; 1728)$

: $2465\; =\; 5\; \backslash cdot\; 17\; \backslash cdot\; 29\; \backslash qquad\; (4\; \backslash mid\; 2464;\backslash quad\; 16\; \backslash mid\; 2464;\backslash quad\; 28\; \backslash mid\; 2464)$

: $2821\; =\; 7\; \backslash cdot\; 13\; \backslash cdot\; 31\; \backslash qquad\; (6\; \backslash mid\; 2820;\backslash quad\; 12\; \backslash mid\; 2820;\backslash quad\; 30\; \backslash mid\; 2820)$

: $6601\; =\; 7\; \backslash cdot\; 23\; \backslash cdot\; 41\; \backslash qquad\; (6\; \backslash mid\; 6600;\backslash quad\; 22\; \backslash mid\; 6600;\backslash quad\; 40\; \backslash mid\; 6600)$

: $8911\; =\; 7\; \backslash cdot\; 19\; \backslash cdot\; 67\; \backslash qquad\; (6\; \backslash mid\; 8910;\backslash quad\; 18\; \backslash mid\; 8910;\backslash quad\; 66\; \backslash mid\; 8910).$

In 1910, Carmichael himself{{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 }} also published the smallest such number, 561, and the numbers were later named after him.

Jack Chernick{{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 }} proved a theorem in 1939 which can be used to construct a subset of Carmichael numbers. The number $(6k\; +\; 1)(12k\; +\; 1)(18k\; +\; 1)$ 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 $n$, there are at least $n^\{2/7\}$ Carmichael numbers between 1 and {{tmath|1= n }}.

Thomas Wright proved that if $a$ and $m$ 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 }}.{{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 }}

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,{{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}} so the largest known Carmichael number is much greater than the largest known prime.

Carmichael numbers have at least three positive prime factors. The first Carmichael numbers with $k\; =\; 3,\; 4,\; 5,\; \backslash ldots$ prime factors are {{OEIS|id=A006931}}:

The first Carmichael numbers with 4 prime factors are {{OEIS|id=A074379}}:

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) is the Hardy-Ramanujan Number: the smallest number that can be expressed as the sum of two cubes (of positive numbers) in two different ways.

Let $C(X)$ denote the number of Carmichael numbers less than or equal to {{tmath|1= X }}. The distribution of Carmichael numbers by powers of 10 {{OEIS|id=A055553}}:

In 1953, Knödel proved the upper bound:

:

for some constant {{tmath|1= k_1 }}.

In 1956, Erdős improved the bound to

:

for some constant {{tmath|1= k_2 }}.{{cite journal |author=Erdős, P. |year=2022 |title=On pseudoprimes and Carmichael numbers |journal=Publ. Math. Debrecen |volume=4 |issue=3–4 |pages=201–206 |doi=10.5486/PMD.1956.4.3-4.16 |url=http://www.renyi.hu/~p_erdos/1956-10.pdf |archive-url=https://web.archive.org/web/20110611160632/http://www.renyi.hu/~p_erdos/1956-10.pdf |archive-date=2011-06-11 |url-status=live |mr=79031 |s2cid=253789521 |author-link=Paul Erdős }} He further gave a heuristic argument suggesting that this upper bound should be close to the true growth rate of {{tmath|1= C(X) }}.

In the other direction, Alford, Granville and Pomerance proved in 1994 that for sufficiently large X,

: $C(X)\; >\; X^\backslash frac\{2\}\{7\}.$

In 2005, this bound was further improved by Harman{{cite journal |author=Glyn Harman |title=On the number of Carmichael numbers up to x |journal=Bulletin of the London Mathematical Society |volume=37 |issue=5 |year=2005 |pages=641–650 |doi=10.1112/S0024609305004686|s2cid=124405969 }} to

: $C(X)\; >\; X^\{0.332\}$

who subsequently improved the exponent to {{tmath|1= 0.7039 \cdot 0.4736 = 0.33336704 > 1/3 }}.

{{cite journal |last=Harman |first=Glyn

|date=2008 |title=Watt's mean value theorem and Carmichael numbers

|doi=10.1142/S1793042108001316 |mr=2404800

|journal=International Journal of Number Theory |volume=4 |issue=2 |pages=241–248

}}

Regarding the asymptotic distribution of Carmichael numbers, there have been several conjectures. In 1956, Erdős conjectured that there were $X^\{1-o(1)\}$ Carmichael numbers for X sufficiently large. In 1981, Pomerance{{cite journal |author=Pomerance, C. |year=1981 |title=On the distribution of pseudoprimes |journal=Math. Comp. |volume=37 |issue=156 |pages=587–593|jstor=2007448 |doi=10.1090/s0025-5718-1981-0628717-0|author-link=Carl Pomerance |doi-access=free }} sharpened Erdős' heuristic arguments to conjecture that there are at least

: $X\; \backslash cdot\; L(X)^\{-1\; +\; o(1)\}$

Carmichael numbers up to {{tmath|1= X }}, where {{tmath|1= L(x) = \exp{ \left( \frac{\log x \log\log\log x} {\log\log x} \right) } }}.

However, inside current computational ranges (such as the count of Carmichael numbers performed by Goutier{{OEIS|id=A055553}} up to 10²²), these conjectures are not yet borne out by the data; empirically, the exponent is $C(X)\; \backslash approx\; X^\{0.35\}$ for the highest available count (C(X)=49679870 for X= 10²²).

In 2021, Daniel Larsen proved an analogue of Bertrand's postulate for Carmichael numbers first conjectured by Alford, Granville, and Pomerance in 1994.{{cite journal |author=Larsen, Daniel |date=20 July 2022 |title=Bertrand's Postulate for Carmichael Numbers |url=https://academic.oup.com/imrn/advance-article-abstract/doi/10.1093/imrn/rnac203/6647493 |journal=International Mathematics Research Notices |volume=2023 |issue=15 |pages=13072–13098 |arxiv=2111.06963 |doi=10.1093/imrn/rnac203}} Using techniques developed by Yitang Zhang and James Maynard to establish results concerning small gaps between primes, his work yielded the much stronger statement that, for any $\backslash delta>0$ and sufficiently large $x$ in terms of $\backslash delta$, there will always be at least

: $\backslash exp\{\backslash left(\backslash frac\{\backslash log\{x\}\}\{(\backslash log\; \backslash log\{x\})^\{2+\backslash delta\}\}\backslash right)\}$

Carmichael numbers between $x$ and

: $x+\backslash frac\{x\}\{(\backslash log\{x\})^\{\backslash frac\{1\}\{2+\backslash delta\}\}\}.$

The notion of Carmichael number generalizes to a Carmichael ideal in any number field {{tmath|1= K }}. For any nonzero prime ideal $\backslash mathfrak\; p$ in {{tmath|1= {\mathcal O}_K }}, we have $\backslash alpha^\{\{\backslash rm\; N\}(\backslash mathfrak\; p)\}\; \backslash equiv\; \backslash alpha\; \backslash bmod\; \{\backslash mathfrak\; p\}$ for all $\backslash alpha$ in {{tmath|1= {\mathcal O}_K }}, where $\{\backslash rm\; N\}(\backslash mathfrak\; p)$ is the norm of the ideal {{tmath|1= \mathfrak p }}. (This generalizes Fermat's little theorem, that $m^p\; \backslash equiv\; m\; \backslash bmod\; p$ for all integers {{tmath|1= m }} when {{tmath|1= p }} is prime.) Call a nonzero ideal $\backslash mathfrak\; a$ in $\{\backslash mathcal\; O\}\_K$ Carmichael if it is not a prime ideal and $\backslash alpha^\{\{\backslash rm\; N\}(\backslash mathfrak\; a)\}\; \backslash equiv\; \backslash alpha\; \backslash bmod\; \{\backslash mathfrak\; a\}$ for all {{tmath|1= \alpha \in {\mathcal O}_K }}, where $\{\backslash rm\; N\}(\backslash mathfrak\; a)$ is the norm of the ideal {{tmath|1= \mathfrak a }}. When {{tmath|1= K }} is {{tmath|1= \mathbf Q }}, the ideal $\backslash mathfrak\; a$ is principal, and if we let {{tmath|1= a }} be its positive generator then the ideal $\backslash mathfrak\; a\; =\; (a)$ is Carmichael exactly when {{tmath|1= a }} is a Carmichael number in the usual sense.

When {{tmath|1= K }} is larger than the rationals it is easy to write down Carmichael ideals in {{tmath|1= {\mathcal O}_K }}: for any prime number {{tmath|1= p }} that splits completely in {{tmath|1= K }}, the principal ideal $p\{\backslash mathcal\; O\}\_K$ is a Carmichael ideal. Since infinitely many prime numbers split completely in any number field, there are infinitely many Carmichael ideals in {{tmath|1= {\mathcal O}_K }}. For example, if {{tmath|1= p }} is any prime number that is 1 mod 4, the ideal {{tmath|1= (p) }} in the Gaussian integers $\backslash mathbb\; Z[i]$ is a Carmichael ideal.

Both prime and Carmichael numbers satisfy the following equality:

: $\backslash gcd\; \backslash left(\backslash sum\_\{x=1\}^\{n-1\}\; x^\{n-1\},\; n\backslash right)\; =\; 1.$

{{main|Lucas–Carmichael number}}

A positive composite integer $n$ is a Lucas–Carmichael number if and only if $n$ is square-free, and for all prime divisors $p$ of {{tmath|1= n }}, it is true that {{tmath|1= p + 1 \mid n + 1 }}. The first Lucas–Carmichael numbers are:

: 399, 935, 2015, 2915, 4991, 5719, 7055, 8855, 12719, 18095, 20705, 20999, 22847, 29315, 31535, 46079, 51359, 60059, 63503, 67199, 73535, 76751, 80189, 81719, 88559, 90287, ... {{OEIS|A006972}}

Quasi–Carmichael numbers are squarefree composite numbers {{tmath|1= n }} with the property that for every prime factor {{tmath|1= p }} of {{tmath|1= n }}, {{tmath|1= p + b }} divides {{tmath|1= n + b }} positively with {{tmath|1= b }} being any integer besides 0. If {{tmath|1= b = -1 }}, these are Carmichael numbers, and if {{tmath|1= b = 1}}, these are Lucas–Carmichael numbers. The first Quasi–Carmichael numbers are:

: 35, 77, 143, 165, 187, 209, 221, 231, 247, 273, 299, 323, 357, 391, 399, 437, 493, 527, 561, 589, 598, 713, 715, 899, 935, 943, 989, 1015, 1073, 1105, 1147, 1189, 1247, 1271, 1295, 1333, 1517, 1537, 1547, 1591, 1595, 1705, 1729, ... {{OEIS|A257750}}

{{main|Knödel number}}

An n-Knödel number for a given positive integer n is a composite number m with the property that each {{tmath|1= i < m }} coprime to m satisfies {{tmath|1= i^{m - n} \equiv 1 \pmod{m} }}. The {{tmath|1= n = 1 }} case are Carmichael numbers.

Carmichael numbers can be generalized using concepts of abstract algebra.

The above definition states that a composite integer n is Carmichael

precisely when the nth-power-raising function p_n from the ring Z_n of integers modulo n to itself is the identity function. The identity is the only Z_n-algebra endomorphism on Z_n so we can restate the definition as asking that p_n be an algebra endomorphism of Z_n.

As above, p_n satisfies the same property whenever n is prime.

The nth-power-raising function p_n is also defined on any Z_n-algebra A. A theorem states that n is prime if and only if all such functions p_n are algebra endomorphisms.

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_n is an endomorphism on every Z_n-algebra that can be generated as Z_n-module by m elements. Carmichael numbers of order 1 are just the ordinary Carmichael numbers.

According to Howe, 17 · 31 · 41 · 43 · 89 · 97 · 167 · 331 is an order 2 Carmichael number. This product is equal to 443,372,888,629,441.{{cite journal |author = Everett W. Howe |title=Higher-order Carmichael numbers |journal=Mathematics of Computation |date=October 2000 |volume=69 |issue=232 |pages=1711–1719 |arxiv=math.NT/9812089 |jstor=2585091 |doi=10.1090/s0025-5718-00-01225-4|bibcode=2000MaCom..69.1711H |s2cid=6102830 }}

Korselt's criterion can be generalized to higher-order Carmichael numbers, as shown by Howe.

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.

{{reflist|40em}}

• {{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=https://www.ams.org/journals/bull/1910-16-05/home.html |doi=10.1090/s0002-9904-1910-01892-9|doi-access=free }}
• {{cite journal |author=Carmichael, R. D. |year=1912 |title=On composite numbers P which satisfy the Fermat congruence $a^\{P-1\}\backslash equiv\; 1\backslash bmod\; P$ |journal=American Mathematical Monthly |volume=19 |issue=2 |pages=22–27 |doi=10.2307/2972687|jstor=2972687 }}
• {{cite journal |author=Chernick, J. |year=1939 |title=On Fermat's simple theorem |journal=Bull. Amer. Math. Soc. |volume=45 |issue=4 |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 }}
• {{cite journal |author=Korselt, A. R. |year=1899 |title=Problème chinois |journal=L'Intermédiaire des Mathématiciens |volume=6 |pages=142–143}}
• {{cite journal |author1=Löh, G. |author2=Niebuhr, W. |year=1996 |url=https://www.ams.org/mcom/1996-65-214/S0025-5718-96-00692-8/S0025-5718-96-00692-8.pdf |archive-url=https://web.archive.org/web/20030425040718/http://www.ams.org/mcom/1996-65-214/S0025-5718-96-00692-8/S0025-5718-96-00692-8.pdf |archive-date=2003-04-25 |url-status=live |title=A new algorithm for constructing large Carmichael numbers |journal=Math. Comp. |volume=65 |issue=214 |pages=823–836 |doi=10.1090/S0025-5718-96-00692-8|bibcode=1996MaCom..65..823L |doi-access=free }}
• {{cite book | title = The Book of Prime Number Records | publisher = Springer | year = 1989 | isbn = 978-0-387-97042-4 | author = Ribenboim, P. | author-link = Paulo Ribenboim }}
• {{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 |doi=10.21136/CPMF.1885.122245 |url=http://dml.cz/handle/10338.dmlcz/122245|doi-access=free }}

• {{springer|title=Carmichael number|id=p/c110100}}
• [https://encyclopediaofmath.org/wiki/Carmichael_number Encyclopedia of Mathematics]
• [https://web.archive.org/web/20090906105152/http://de.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Carmichael-Zahlen Table of Carmichael numbers]
• [https://github.com/drazioti/Carmichael/tree/master/Tables Tables of Carmichael numbers with many prime factors]
• [http://www.s369624816.websitehome.co.uk/rgep/cartable.html Tables of Carmichael numbers below $10^\{18\}$]
• {{MathPages|id=home/kmath028/kmath028|title=The Dullness of 1729}}
• {{MathWorld | urlname=CarmichaelNumber | title=Carmichael Number}}
• [http://www.numericana.com/answer/modular.htm Final Answers Modular Arithmetic]

{{Classes of natural numbers}}

Category:Eponymous numbers in mathematics

Category:Integer sequences

Category:Modular arithmetic

Category:Pseudoprimes