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an ≡ a (mod n) (see modular arithmetic).
Carmichael numbers are important because they can fool the
Fermat primality test, thus they are always fermat liars. If Carmichael numbers did not exist, this primality test could always be used to prove compositeness of a number.Even though as numbers become larger Carmichael numbers become rarer, the sheer number of them makes the Fermat primality test largely useless compared to other primality tests such as the Solovay-Strassen primality test. For example, there are 105,212 Carmichael numbers between 1 and 1015.
An alternative and equivalent definition of Carmichael numbers is given by Korselt's theorem from 1899.
Theorem (Korselt 1899): 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, it is true that p − 1 divides n − 1.
It follows from this theorem that all Carmichael numbers are odd.
Korselt was the first who observed these properties, but he could not find an example. In 1910 Robert Daniel Carmichael found the first and smallest such number, 561, and hence the name.
That 561 is a Carmichael number can be seen with Korselt's theorem. Indeed, 561 = 3 ˇ 11 ˇ 17 is squarefree and 2 | 560, 10 | 560 and 16 | 560. (The notation a | b means: a dividesNumber theory In mathematics, a divisor of an integer n also called a factor of n is an integer which evenly divides n without leaving a remainder. For example, 7 is a divisor of 42 because 42/7 6. We also say 42 is divisible by 7 or 42 is a multiple of 7 b).
The next few Carmichael numbers are (sequence 002997 in OEIS):
J. Chernick 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.
Paul Erdös heuristically argued there should be infinitely many Carmichael numbers. In 19941994 is a common year starting on Saturday, and was designated the International year of the Family''. Events January events January 1 North American Free Trade Agreement (NAFTA) goes into effect January 6 Nancy Kerrigan is clubbed on the right leg by an it was shown by William Alford, Andrew Granville and Carl Pomerance that there really exist infinitely many Carmichael numbers.It has also been shown that for sufficiently large n, there are at least n2/7 Carmichael numbers between 1 and n.
Carmichael numbers have at least three positive prime factors. The first Carmichael numbers with k = 3, 4, 5, … prime factors are (sequence A006931 in OEIS):
| k | |
|---|---|
| 3 | 561 = 3 ˇ 11 ˇ 17 |
| 4 | 41041 = 7 ˇ 11 ˇ 13 ˇ 41 |
| 5 | 825265 = 5 ˇ 7 ˇ 17 ˇ 19 ˇ 73 |
| 6 | 321197185 = 5 ˇ 19 ˇ 23 ˇ 29 ˇ 37 ˇ 137 |
| 7 | 5394826801 = 7 ˇ 13 ˇ 17 ˇ 23 ˇ 31 ˇ 67 ˇ 73 |
| 8 | 232250619601 = 7 ˇ 11 ˇ 13 ˇ 17 ˇ 31 ˇ 37 ˇ 73 ˇ 163 |
| 9 | 9746347772161 = 7 ˇ 11 ˇ 13 ˇ 17 ˇ 19 ˇ 31 ˇ 37 ˇ 41 ˇ 641 |
The first Carmichael numbers with 4 prime factors are (sequence A074379 in OEIS):
| i | |
|---|---|
| 1 | 41041 = 7 ˇ 11 ˇ 13 ˇ 41 |
| 2 | 62745 = 3 ˇ 5 ˇ 47 ˇ 89 |
| 3 | 63973 = 7 ˇ 13 ˇ 19 ˇ 37 |
| 4 | 75361 = 11 ˇ 13 ˇ 17 ˇ 31 |
| 5 | 101101 = 7 ˇ 11 ˇ 13 ˇ 101 |
| 6 | 126217 = 7 ˇ 13 ˇ 19 ˇ 73 |
| 7 | 172081 = 7 ˇ 13 ˇ 31 ˇ 61 |
| 8 | 188461 = 7 ˇ 13 ˇ 19 ˇ 109 |
| 9 | 278545 = 5 ˇ 17 ˇ 29 ˇ 113 |
| 10 | 340561 = 13 ˇ 17 ˇ 23 ˇ 67 |
Incidentally, the first Carmichael number (561) is expressible as the sum of two first powers in more ways than any smaller number (although admittedly all numbers share this property). 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 Number199e03 1729 This article is about the number 1729. For the year AD 1729, see 1729. 1729 is known as the Hardy-Ramanujan number after a famous anecdote of the British mathematician G. Hardy regarding a hospital visit to the Indian mathematician Srinivasa R: the smallest number that can be expressed as the sum of two cubes in two different ways.
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