## Generalizations of Fermat's little theorem in
rings with a multiplicative identity

### Anthony Narkawicz

Fermat's Little Theorem states that if *p* is a prime
number that does
not divide *a*, then *a*^{(p-1)} __=__
*a* mod *p*. Similar theorems include
Euler's Theorem, which states that if *a* and *m*
are relatively prime,
then *a*^{m} __=__ 1 mod *m*. These theorems
are especially applicable
in basic coding methods such as RSA encryption. As a source of
motivation, this paper aims in part to answer the following question
(a generalization of FLT): What are necessary and sufficient
conditions on *Q* and *m* so that
*a*^{Q} __=__ *a* mod *m* for every
integer *a*?
This question is easily answered, but more generally, we are able to
answer the following question. Let *R* be any ring with a 1,
let *Q* be an integer at least 2, and let *I* be an ideal
of *R*. What are necessary
and sufficient conditions on *Q* and *I* so that
*a*^{Q} __=__ *a* mod *I* for
every *a* in *R*? The result is that *a*^{Q}
__=__ *a* mod *I* for every *a* in *R* if
and only if *I* is the intersection of all maximal ideals
containing *I*, and for any such maximal ideal *M*,
we have that *R/M* is a field and |*R/M*|-1
divides *Q*-1.