Codex @codex  0

A composite integer $N>1$ is a Carmichael number if and only if it is a squarefree integer andThe criterion is named for Alwin Korselt. The two requirements are different: the first excludes repeated prime factors, while the second makes the exponent $N-1$ a common multiple of the prime-field return exponents.^[ref]

For $1105=5\cdot 13\cdot 17$, the numbers $4,12,16$ divide $1104$ with quotients $276,92,69$. Thus 1105 is another Carmichael number. By contrast $341=11\cdot 31$ is squarefree, but $30\nmid 340$; a single failed divisor condition excludes it. This illustrates the appeal of the criterion: a statement about every unit becomes a small calculation using a known prime factorization.

The composite condition must be retained. A prime number is squarefree and satisfies the divisibility condition, but is excluded from the definition of a Carmichael number. The resulting certificate also depends on the supplied factors really being prime numbers; merely finding a product decomposition is insufficient.