Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-16T17:17:34.328Z Has data issue: false hasContentIssue false
  • English
  • Français

Finding Carmichael numbers

Published online by Cambridge University Press:  23 January 2015

G. J. O. Jameson
G. J. O. Jameson*
Affiliation:
Dept. of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF e-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

Recall that Fermat's ‘little theorem’ says that if p is prime and a is not a multiple of p, then ap − 1 ≡ 1 mod p.

This theorem gives a possible way to detect primes, or more exactly, non-primes: if for some positive a ≤ n − 1, an − 1 is not congruent to 1 mod n, then, by the theorem, n is not prime. A lot of composite numbers can indeed be detected by this test, but there are some that evade it. In other words, there are numbers n that are composite but still satisfy an − 1 ≡ 1 mod n for all a coprime to n. Such numbers might be called ‘false primes’, but in fact they are called Carmichael numbers in honour of R. D. Carmichael, who demonstrated their existence in 1912 [1] – so the year 2012 marks their centenary. (Composite numbers that satisfy the stated condition for one particular a are called a-pseudoprimes. They are the subject of a companion article [2].)

Type
Articles
Information
The Mathematical Gazette , Volume 95 , Issue 533 , July 2011 , pp. 244 - 255
Copyright
Copyright © The Mathematical Association 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Carmichael, R. D., On composite numbers P which satisfy the Fermat congruence ap − 1 ≡ 1 (mod P), American Math. Monthly 19 (1912), pp.22–27.Google Scholar
2. Jameson, G. J. O., Finding pseudoprimes (to appear in a future Gazette).Google Scholar
3. Jones, G. A. and Jones, J. M., Elementary number theory, Springer (1998).Google Scholar
4. Pinch, R. G. E., The Carmichael numbers up to 1015 , Math. Compo 61 (1993) pp. 381–391.Google Scholar
5. Pinch, R. G. E., The Carmichael numbers up to 1018, at www.chalcedon.demon.co.uk/rgep/carpsp.html Google Scholar
6. Ribenboim, P., The new book of prime number records, Springer (1995).Google Scholar
7. Jameson, G. J. O., Carmichael numbers with three prime factors, at www.maths.lancs.ac.uk/~jameson Google Scholar
8. Beeger, N. G. W. H., On composite numbers n for which an − 1 ≡ 1 mod n for every a prime to n , Scripta Math. 16 (1950) pp. 133–135.Google Scholar
9. Duparc, H. J. A., On Carmichael numbers, Simon Stevin 29 (1952) pp.21–24.Google Scholar
10. Alford, W. R, Granville, Andrew and Pomerance, Carl, There are infinitely many Carmichael numbers, Annals of Math. 140 (1994) pp. 703–722.Google Scholar
11. Harman, Glyn, On the number of Carmichael numbers up to x , Bull. London Math. Soc. 37 (2005) pp. 641–650.Google Scholar
12. Erdos, P., On pseudoprimes and Carmichael numbers, Pub. Math. Debrecen 4 (1956) pp. 200–206.Google Scholar
13. Pomerance, Carl, Two methods in elementary number theory, number theory and applications (Banff, 1988; RA. Mollin, ed.), NATO Adv. Sci. Ser. C 265 (1989) pp. 135–161.Google Scholar
14. Granville, Andrew and Pomerance, Carl, Two contradictory conjectures concerning Carmichael numbers,Math. Compo 71 (2001) pp. 883–908.Google Scholar
15. Damgard, I., Landrock, P. and Pomerance, C., Average case error estimates for the strong probable prime test, Math. Compo 61 (1993) pp.177–194.Google Scholar
16. Balasubramanian, R and Nagaraj, S. V., Density of Carmichael nutnbers with three prime factors, Math. Compo 66 (1997) pp. 1705–1708.Google Scholar
17. Heath-Brown, D. R., Carmichael numbers with three prime factors, Hardy-Ramanujan J. 30 (2007) pp. 6–12.Google Scholar