Carmichael number

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Template:Pseudoprimality property

Definition

A composite number is termed an Carmichael number or absolute pseudoprime if it satisfies the following equivalent conditions:

  1. The universal exponent (also called the Carmichael function) of divides .
  2. For any natural number relatively prime to , divides .
  3. is a Fermat pseudoprime to any base relatively prime to it.
  4. is a square-free odd number greater than 1 and divides for every prime divisor of .

Occurrence

Initial examples

561, 1105, 1729, 2465, 2821, 6601, [SHOW MORE]View list on OEIS

Note that Carmichael number is square-free and Carmichael number is odd, so each of these is the product of distinct odd primes. Further, because Carmichael number is not semiprime, there are at least three prime factors of each number. For the first few examples, we indicate the prime factors:

Carmichael number Prime factors as list 3? 5? 7? 11? 13? 17? 19? 23? 29? 31? Universal exponent (must divide number minus one)
561 3, 11, 17 Yes No No Yes No Yes No No No No 80
1105 5, 13, 17 No Yes No No Yes Yes No No No No 48
1729 7, 13, 19 No No Yes No Yes No Yes No No No 36
2465 5, 17, 29 No Yes No No No Yes No No Yes No 112
2821 7, 13, 31 No No Yes No Yes No No No No Yes 60

Facts