Korselt's criterion states that, for an integer 
,
 |
For composite 
, the left-hand condition is equivalent to 
being a Carmichael number,
so these solutions are precisely the Carmichael
numbers (Korselt 1899).
To prove sufficiency, consider each prime divisor 
of 
. If 
, then 
. Otherwise, Fermat's
little theorem and 
give 
, so again 
. The squarefree condition
then combines these divisibilities to give 
.
Conversely, suppose 
for every integer 
. If 
, the Chinese remainder
theorem gives an 
that is congruent to 
modulo 
and to 0 modulo the largest divisor
of 
relatively prime to 
. Then 
, contradicting 
, so 
is squarefree. For a prime divisor 
of 
, use the Chinese remainder
theorem to choose 
congruent to a primitive
root modulo 
and to 1 modulo 
.
The number 
is relatively prime to 
, and 
. Since 
has multiplicative order 
modulo 
, it follows that 
.
