Quote:
Originally Posted by devarajkandadai
Modified Fermat's theorem: Let a belong to the ring of Gaussian integers. Then
a^(p^21) = = 1 (mod p). Here p is a prime number with shape 4m + 1 or 4m + 3.

I assume you intend a and p to be coprime.
The case where p=4m+3 for some integer m is the one we already looked at in your earlier thread:
http://www.mersenneforum.org/showthread.php?t=22223
Suppose \(p=4m+1\) for some integer \(m\).
Then the prime factorization of \(p\) in the Gaussian integers has the form \(p=q\bar{q}\) for some Gaussian prime \(q\) by Theorem 62 in our course,
where \(\bar{q}\) is the complex conjugate of \(q\), which is not an associate of \(q\).
By the Chinese Remainder Theorem, we get
\[ \mathbb{Z}[i]/p\mathbb{Z}[i]\cong\mathbb{Z}[i]/q\mathbb{Z}[i]\times\mathbb{Z}[i]/\bar{q}\mathbb{Z}[i] \]
so also
\[ \left(\mathbb{Z}[i]/p\mathbb{Z}[i]\right)^*\cong\left(\mathbb{Z}[i]/q\mathbb{Z}[i]\right)^*\times\left(\mathbb{Z}[i]/\bar{q}\mathbb{Z}[i]\right)^* \]
Now \(N(q)N(\bar{q})=N(p)=p^2\) so \(N(q)=N(\bar{q})=p\) and therefore \(\left(\mathbb{Z}[i]/q\mathbb{Z}[i]\right)^*\) has \(p1\) elements, and similarly for \(\left(\mathbb{Z}[i]/\bar{q}\mathbb{Z}[i]\right)^*\).
It follows that the order of any element of \(\left(\mathbb{Z}[i]/p\mathbb{Z}[i]\right)^*\) divides \(p1\), which is a factor of \(p^21\), leading to the statement you gave.