for all integers which are relatively prime to . Carmichael numbers are named after American mathematician Robert Carmichael, the term having been introduced by Nicolaas Beeger in 1950 (Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "F numbers" for short). They are infinite in number.
The Carmichael numbers form the subset K1 of the Knödel numbers.
Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . 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 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 so, in theory, either an Euler or a strong probable prime test could prove that a Carmichael number is, in fact, composite.
Arnault gives a 397-digit Carmichael number that is a strong pseudoprime to all prime bases less than 307:
is a 131-digit prime. is the smallest prime factor of , so this Carmichael number is also a (not necessarily strong) pseudoprime to all bases less than .
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).
An alternative and equivalent definition of Carmichael numbers is given by Korselt's criterion.
- Theorem (A. Korselt 1899): A positive composite integer is a Carmichael number if and only if is square-free, and for all prime divisors of , it is true that .
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 results in an even dividing an odd, a contradiction. (The oddness of Carmichael numbers also follows from the fact that is a Fermat witness for any even composite number.) From the criterion it also follows that Carmichael numbers are cyclic. Additionally, it follows that there are no Carmichael numbers with exactly two prime divisors.
Korselt was the first who observed the basic properties of Carmichael numbers, but he did not give any examples. In 1910, Carmichael found the first and smallest such number, 561, which explains the name "Carmichael number".
That 561 is a Carmichael number can be seen with Korselt's criterion. Indeed, is square-free and , and .
These first seven Carmichael numbers, from 561 to 8911, were all found by the Czech mathematician Václav Šimerka in 1885 (thus preceding not just Carmichael but also Korselt, although Šimerka did not find anything like Korselt's criterion). His work, however, remained unnoticed.
J. Chernick proved a theorem in 1939 which can be used to construct a subset of Carmichael numbers. The number 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 , there are at least Carmichael numbers between 1 and 
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, so the largest known Carmichael number is much greater than the largest known prime.
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.
for some constant .
In 1956, Erdős improved the bound to
who subsequently improved the exponent to . 
Regarding the asymptotic distribution of Carmichael numbers, there have been several conjectures. In 1956, Erdős conjectured that there were Carmichael numbers for X sufficiently large. In 1981, Pomerance sharpened Erdős' heuristic arguments to conjecture that there are at least
Carmichael numbers up to , where .
However, inside current computational ranges (such as the counts of Carmichael numbers performed by Pinch up to 1021), these conjectures are not yet borne out by the data.
In 2021, Daniel Larsen, a 17-year-old high-school student from Indiana, proved an analogue of Bertrand's postulate for Carmichael numbers first conjectured by Alford, Granville, and Pomerance in 1994. 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 and sufficiently large in terms of , there will always be at least
Carmichael numbers between and
The notion of Carmichael number generalizes to a Carmichael ideal in any number field K. For any nonzero prime ideal in , we have for all in , where is the norm of the ideal . (This generalizes Fermat's little theorem, that for all integers m when p is prime.) Call a nonzero ideal in Carmichael if it is not a prime ideal and for all , where is the norm of the ideal . When K is , the ideal is principal, and if we let a be its positive generator then the ideal is Carmichael exactly when a is a Carmichael number in the usual sense.
When K is larger than the rationals it is easy to write down Carmichael ideals in : for any prime number p that splits completely in K, the principal ideal is a Carmichael ideal. Since infinitely many prime numbers split completely in any number field, there are infinitely many Carmichael ideals in . 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.
Both prime and Carmichael numbers satisfy the following equality:
- 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, ... (sequence A006972 in the OEIS)
Quasi–Carmichael numbers are squarefree composite numbers n with the property that for every prime factor p of n, p + b divides n + b positively with b being any integer besides 0. If b = −1, these are Carmichael numbers, and if 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, ... (sequence A257750 in the OEIS)
Higher-order 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 pn from the ring Zn of integers modulo n to itself is the identity function. The identity is the only Zn-algebra endomorphism on Zn so we can restate the definition as asking that pn be an algebra endomorphism of Zn. As above, pn satisfies the same property whenever n is prime.
The nth-power-raising function pn is also defined on any Zn-algebra A. A theorem states that n is prime if and only if all such functions pn 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 pn is an endomorphism on every Zn-algebra that can be generated as Zn-module by m elements. Carmichael numbers of order 1 are just the ordinary Carmichael numbers.
An order 2 Carmichael number
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.
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.
- Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. Vol. 126 (second ed.). Boston, MA: Birkhäuser. ISBN 978-0-8176-3743-9. Zbl 0821.11001.
- Crandall, Richard; Pomerance, Carl (2005). Prime Numbers: A Computational Perspective (second ed.). New York: Springer. p. 133. ISBN 978-0387-25282-7.
- Ore, Øystein (1948). Number Theory and Its History. New York: McGraw-Hill. pp. 331–332 – via Internet Archive.
- W. R. Alford; Andrew Granville; Carl Pomerance (1994). "There are Infinitely Many Carmichael Numbers" (PDF). Annals of Mathematics. 140 (3): 703–722. doi:10.2307/2118576. JSTOR 2118576. Archived (PDF) from the original on 2005-03-04.
- Cepelewicz, Jordana (13 October 2022). "Teenager Solves Stubborn Riddle About Prime Number Look-Alikes". Quanta Magazine. Retrieved 13 October 2022.
- D. H. Lehmer (1976). "Strong Carmichael numbers". J. Austral. Math. Soc. 21 (4): 508–510. doi:10.1017/s1446788700019364. 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.
- F. Arnault (August 1995). "Constructing Carmichael Numbers Which Are Strong Pseudoprimes to Several Bases". Journal of Symbolic Computation. 20 (2): 151–161. doi:10.1006/jsco.1995.1042.
- Pinch, Richard (December 2007). Anne-Maria Ernvall-Hytönen (ed.). The Carmichael numbers up to 1021 (PDF). Proceedings of Conference on Algorithmic Number Theory. Vol. 46. Turku, Finland: Turku Centre for Computer Science. pp. 129–131. Retrieved 2017-06-26.
- Carmichael Multiples of Odd Cyclic Numbers "Any divisor of a Carmichael number must be an odd cyclic number"
- Proof sketch: If is square-free but not cyclic, for two prime factors and of . But if satisfies Korselt then , so by transitivity of the "divides" relation . But is also a factor of , a contradiction.
- R. D. Carmichael (1910). "Note on a new number theory function". Bulletin of the American Mathematical Society. 16 (5): 232–238. doi:10.1090/s0002-9904-1910-01892-9.
- V. Šimerka (1885). "Zbytky z arithmetické posloupnosti (On the remainders of an arithmetic progression)". Časopis Pro Pěstování Matematiky a Fysiky. 14 (5): 221–225. doi:10.21136/CPMF.1885.122245.
- Lemmermeyer, F. (2013). "Václav Šimerka: quadratic forms and factorization". LMS Journal of Computation and Mathematics. 16: 118–129. doi:10.1112/S1461157013000065.
- Chernick, J. (1939). "On Fermat's simple theorem" (PDF). Bull. Amer. Math. Soc. 45 (4): 269–274. doi:10.1090/S0002-9904-1939-06953-X.
- Thomas Wright (2013). "Infinitely many Carmichael Numbers in Arithmetic Progressions". Bull. London Math. Soc. 45 (5): 943–952. arXiv:1212.5850. doi:10.1112/blms/bdt013. S2CID 119126065.
- W.R. Alford; et al. (2014). "Constructing Carmichael numbers through improved subset-product algorithms". Math. Comp. 83 (286): 899–915. arXiv:1203.6664. doi:10.1090/S0025-5718-2013-02737-8. S2CID 35535110.
- Erdős, P. (2022). "On pseudoprimes and Carmichael numbers" (PDF). Publ. Math. Debrecen. 4 (3–4): 201–206. doi:10.5486/PMD.1956.4.3-4.16. MR 0079031. S2CID 253789521. Archived (PDF) from the original on 2011-06-11.
- Glyn Harman (2005). "On the number of Carmichael numbers up to x". Bulletin of the London Mathematical Society. 37 (5): 641–650. doi:10.1112/S0024609305004686. S2CID 124405969.
- Harman, Glyn (2008). "Watt's mean value theorem and Carmichael numbers". International Journal of Number Theory. 4 (2): 241–248. doi:10.1142/S1793042108001316. MR 2404800.
- Pomerance, C. (1981). "On the distribution of pseudoprimes". Math. Comp. 37 (156): 587–593. doi:10.1090/s0025-5718-1981-0628717-0. JSTOR 2007448.
- Larsen, Daniel (20 July 2022). "Bertrand's Postulate for Carmichael Numbers". International Mathematics Research Notices. arXiv:2111.06963. doi:10.1093/imrn/rnac203.
- Everett W. Howe (October 2000). "Higher-order Carmichael numbers". Mathematics of Computation. 69 (232): 1711–1719. arXiv:math.NT/9812089. Bibcode:2000MaCom..69.1711H. doi:10.1090/s0025-5718-00-01225-4. JSTOR 2585091. S2CID 6102830.
- Carmichael, R. D. (1910). "Note on a new number theory function". Bulletin of the American Mathematical Society. 16 (5): 232–238. doi:10.1090/s0002-9904-1910-01892-9.
- Carmichael, R. D. (1912). "On composite numbers P which satisfy the Fermat congruence ". American Mathematical Monthly. 19 (2): 22–27. doi:10.2307/2972687. JSTOR 2972687.
- Chernick, J. (1939). "On Fermat's simple theorem" (PDF). Bull. Amer. Math. Soc. 45 (4): 269–274. doi:10.1090/S0002-9904-1939-06953-X.
- Korselt, A. R. (1899). "Problème chinois". L'Intermédiaire des Mathématiciens. 6: 142–143.
- Löh, G.; Niebuhr, W. (1996). "A new algorithm for constructing large Carmichael numbers" (PDF). Math. Comp. 65 (214): 823–836. Bibcode:1996MaCom..65..823L. doi:10.1090/S0025-5718-96-00692-8. Archived (PDF) from the original on 2003-04-25.
- Ribenboim, P. (1989). The Book of Prime Number Records. Springer. ISBN 978-0-387-97042-4.
- Šimerka, V. (1885). "Zbytky z arithmetické posloupnosti (On the remainders of an arithmetic progression)". Časopis Pro Pěstování Matematiky a Fysiky. 14 (5): 221–225. doi:10.21136/CPMF.1885.122245.
- "Carmichael number", Encyclopedia of Mathematics, EMS Press, 2001 
- Encyclopedia of Mathematics
- Table of Carmichael numbers
- Tables of Carmichael numbers with many prime factors
- Tables of Carmichael numbers below
- "The Dullness of 1729". MathPages.com.
- Weisstein, Eric W. "Carmichael Number". MathWorld.
- Final Answers Modular Arithmetic