**Fermat's Little Theorem**: If $p$ is a prime and $\gcd(a,p)=1$ then $a^{p-1} \equiv1\pmod p$.

Over the years, Fermat's Little Theorem have been generalized in several ways. I am aware of four different generalizations as given below.

**1. Euler**: If $\gcd(a,n)=1$ then $a^{\phi(n)} \equiv1 \pmod n$.

**2. Ramachandra**: $\sum_{d|n}\mu(d)a^{n/d} \equiv 0\pmod n$. (*Fermat's Little Theorem follows when $n=p$ is a prime and has only two divisors 1 and $p$*.

**3**. Let $d$ be a divisor of $\phi(n)$. There are exactly $d$ distinct positive integers $r_k, (k=1,2, \ldots d)$ such if $\gcd(x,n)=1$ then $x^{\phi(n)/d} \equiv r_k \pmod n$ for some $(k=1,2, \ldots d)$ (*Euler's generalization itself is a special case of this result when $d=1$*.)

**4. Florentin Smarandache**: $a^{\phi(n_s)+s} \equiv a^s \pmod n$ where $s$ and $n_s$ are defined in Smarandache's paper

I would like to know if there is any other generalization of Fermat's Little Theorem.