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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: g.jameson@lancaster.ac.uk
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Extract

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

This theorem gives a possible way to detect primes, or more exactly, non-primes: if for some positivea ≤ 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].)

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

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References

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