Carmichael number

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A Carmichael number is a composite number named after the mathematician Robert Daniel Carmichael. A Carmichael number divides for every integer . A Carmichael number c also satisfies the congruence , if . The first few Carmichael numbers are 561, 1105, 1729, 2465, 2821, 6601 and 8911. In 1994 Pomerance, Alford and Granville proved that there exist infinitely many Carmichael numbers.

Properties

  • Every Carmichael number is square-free and has at least three different prime factors
  • For every Carmichael number c it holds that is divisible by for every one of its prime factors .
  • Every absolute Euler pseudoprime is a Carmichael number.

Chernick's Carmichael numbers

J. Chernick found in 1939 a way to construct Carmichael numbers[1] [2]. If, for a natural number n, the three numbers , and are prime numbers, the product is a Carmichael number. This condition can only be satisfied if the number ends with 0, 1, 5 or 6. An equivalent formulation of Chernick's construction is that if , and are prime numbers, then the product is a Carmichael number.

This way to construct Carmichael numbers may be extended[3] to

with the condition that each of the factors is prime and that is divisible by .

Distribution of Carmichael numbers

Let C(X) denote the number of Carmichael numbers less than or equal to X. Then for all sufficiently large X

The upper bound is due to Erdős(1956)[4] and Pomerance, Selfridge and Wagstaff (1980)[5] and the lower bound is due to Glyn Harman (2005),[6] improving the earlier lower bound of obtained by Alford, Granville and Pomerance (1994), which first established that there were infinitely many Carmichael numbers.[7]. The asymptotic rate of growth of C(X) is not known.[8]

References and notes

  1. J. Chernick, "On Fermat's simple theorem", Bull. Amer. Math. Soc. 45 (1939) 269-274
  2. (2003-11-22) Generic Carmichael Numbers
  3. Paulo Ribenboim, The new book of prime number records, Springer-Verlag (1996) ISBN 0-387-94457-5. P.120
  4. Paul Erdős, "On pseudoprimes and Carmichael numbers", Publ. Math. Debrecen 4 (1956) 201-206. MR 18 18
  5. C. Pomerance, J.L. Selfridge and S.S. Wagstaff jr, "The pseudoprimes to 25.109", Math. Comp. 35 (1980) 1003-1026. MR 82g:10030
  6. Glyn Harman (2005). "On the number of Carmichael numbers up to x". Bulletin of the London Mathematical Society 37: 641–650. DOI:10.1112/S0024609305004686. Zbl. 1108.11065. Research Blogging.
  7. W. R. Alford, A. Granville, and C. Pomerance (1994). "There are Infinitely Many Carmichael Numbers". Annals of Mathematics 139: 703-722. MR 95k:11114 Zbl 0816.11005.
  8. Richard Guy (2004). Unsolved problems in Number Theory, 3rd. Springer-Verlag. ISBN 0-387-20860-7. . Section A13