Any prime satisfiers this (Fermat's little theorem):
For any integer .
However, the converse fails.
This is an example of a pseudoprime (to base two) for which the converse fails.
A pseudoprime to all base is call a Carmichaeal number.
Since are all primes we have.
Then, Fermat's Elegant theorem (different version than above)
Important Note 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, if .
And, are primes.
is a Carmichael number.
I am supprised, where did you get this theorem from? First time I hear of it. It is really interesting.