Carmichael Numbers
Checks whether a number is a Carmichael number.
carmichael(n)
n |
natural number |
A natural number n is a Carmichael number if it is a Fermat
pseudoprime for every a, that is a^(n-1) = 1 mod n, but
is composite, not prime.
Here the Korselt criterion is used to tell whether a number n is
a Carmichael number.
Returns TRUE or FALSE
There are infinitely many Carmichael numbers, specifically there should
be at least n^(2/7) Carmichael numbers up to n (for n large enough).
R. Crandall and C. Pomerance. Prime Numbers - A Computational Perspective. Second Edition, Springer Science+Business Media, New York 2005.
carmichael(561) # TRUE
## Not run:
for (n in 1:100000)
if (carmichael(n)) cat(n, '\n')
## 561 2821 15841 52633
## 1105 6601 29341 62745
## 1729 8911 41041 63973
## 2465 10585 46657 75361
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.