Notation - meaning of mod(x,y)

Question

Reading about AKS primality test I've encountered the expression $(x+1)^n \equiv x^n+1 \mod{(n, x^r-1)}$. What does the notation $x\equiv y\mod{(r,s)}$ mean in general? Is it simply $x \equiv y \pmod r\pmod s$?

Answer 1

In your example is means modulo the ideal of the polynomial ring $\Bbb Z[x]$ generated by $n$ and $x^r-1$. This ideal can be denoted as $(n,x^r-1)$ which equals $n\Bbb Z[x]+(x^r-1)\Bbb Z[x]$. Your congruence means that $(x+1)^n-(x^n+1)$ is an element of this ideal.