Euler's Theorem states that for any integer $a$ and any integer $n$ such that $a$ and $n$ are coprime (i.e., $\gcd(a, n) = 1$), the following congruence holds:

$a^{\phi(n)} \equiv 1 \ (\text{mod}\ n)$

where $\phi(n)$ is Euler's totient function, which counts the number of integers less than $n$ that are coprime to $n$.

Extended Form

Euler's Theorem is often viewed as a specific case of Fermat's Little Theorem, which is applicable when $n$ is a prime. However, an extended form of Euler's Theorem can be seen when we generalize the formula to a case involving modular exponentiation and the Chinese Remainder Theorem.

Generalized Form (Carmichael's Theorem)

One extended form, often attributed to Carmichael's Function $\lambda(n)$, refines the totient function in some cases. For all $a$ coprime to $n$:

$a^{\lambda(n)} \equiv 1 \ (\text{mod}\ n)$

Where $\lambda(n)$ is the Carmichael function, which is always less than or equal to Euler's totient function, $\phi(n)$, and gives the smallest such exponent.

1. If $n$ is prime, $\lambda(n) = \phi(n)$.
2. For composite $n$, $\lambda(n)$ divides $\phi(n)$ and sometimes provides a smaller exponent.

Practical Use in Cryptography

Euler's Theorem is a key concept in the RSA cryptosystem, where it's used to show that the RSA encryption and decryption process works based on the modular inverse relationship.

Would you like to dive into more examples or the role of Carmichael's function in cryptography?

Here are 5 questions based on the extended information:

1. How does Carmichael’s function differ from Euler’s Totient function?
2. Can Euler's theorem be applied to non-coprime integers?
3. How does Euler’s theorem relate to Fermat's Little Theorem?
4. What is the significance of Euler’s Theorem in RSA encryption?
5. How is the totient function calculated for composite numbers?

Tip: Euler’s Theorem is especially useful in simplifying large modular exponentiations when using numbers that are relatively prime.