What is the significance of i^Q mod P = 1 and are there any special properties to be aware of? I'm studying UProve, and one of the fundamental components of this technology is based upon the relationship between


  i^Q mod P = 1


Lacking a specific name I can call this mathematical relationship, where can I learn more about this equation?

I've heard that not only will the equation equal one, but there are instances it may equal P. 

PS - I'm not sure how to tag this so your feedback is welcome, or let me know if this is better on the math.SE
A special case of the relation you've found is called Fermat's little theorem.
In a more "mathy" way this looks like:

$i^Q\equiv 1 \pmod p$

which will always hold for $Q=p-1$ and for any $Q|p-1$.

The use of this theorem is limited in cryptography, however the problem of determining $Q$, if only $i$ and $p$ are known, this problem is called "discrete logarithm problem" (assuming there isn't a "1" on the right side but rather some random number).

If $p=r*s$ and you're given $Q$ then this is called the RSA-problem (the problem if finding the Q-th root modulo p), again assuming the right side isn't 1 but rather some random number.