First I need to explain what a Carmichael number is.

Any prime satisfiers this (Fermat's little theorem):

For any integer .

However, the converse fails.

For example,

.

This is an example of apseudoprime(to base two) for which the converse fails.

A pseudoprime to all base is call aCarmichaealnumber.

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Since are all primes we have.

Let .

Then, Fermat's Elegant theorem (different version than above)

Thus,

ImportantNote is an integer because primes are odd (usually) and we require for to be an integer we need that .

Now, because of relative primeness in the moduli we have,

Thus,

.

Thus, if .

And, are primes.

Then,

is a Carmichael number.

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I am supprised, where did you get this theorem from? First time I hear of it. It is really interesting.