I have a couple of silly questions (it will definitely demonstrate my lack of ability in mathematics :P)
Is there a type of reduction or absorption of modulus in congruence equations? Here's an example:
If I have $$x (mod 3) \equiv 22 (mod 3)$$, is that the same as finding $ x \equiv 22 mod 3$ or could I get rid of the modulus all together, making $x=22$? I'm trying to prove that the sum of all primitive roots mod $p$ is $\mu(p-1)$ mod $p$.
Secondly, any tips on how to prove the above theorem mentioned (with the sum of all primitive roots)? It took me a while to realize that I can write the sum of all primitive roots as $$\sum_{i=1}^k (a^{r_i})^m$$. I apologize if there's anything obvious I'm missing...I've never taken number theory, and weak in abstract algebra.
There’s a problem in that mathematicians and computer-scientists use “mod” differently. Mathematician says “$-2\equiv13\pmod5$”, meaning that congruence modulo $5$ is an equivalence relation, $a\equiv b\pmod n$ means that $b-a$ is an integer multiple of $n$.
Computer scientist says, “$-2\bmod5=3$”, making “$\!\bmod n$” a function whose range is the set $\{0,1,\cdots,n-1\}$, the value being computed as the (nonnegative) remainder of the input upon Euclidean division by $n$. In this convention, we can say $-2\bmod5=13\bmod5$, because both times, the value of the expression is $3$.
For mathematicians, the computer-science convention is (almost) useless, because it doesn’t generalize to other systems than the integers, even in such a simple case as the Gaussian integers.