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 $x \equiv y \pmod r\pmod s$?

What is $r$ in this context? – Bernard Oct 01 '19 at 19:11 — Bernard, Oct 01 '19 at 19:11

score 4 · Accepted Answer · answered Oct 01 '19 at 19:14

4

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.

answered Oct 01 '19 at 19:14

Angina Seng

158,341

I messed up the notation I copied from wikipedia. See Matthew Daly's copy. Does this answer hold also for the notation $\pmod{x^r-1,n}$? – GottlobtFrege Oct 01 '19 at 19:18
1

@GottlobtFrege That's the same ideal. – Angina Seng Oct 01 '19 at 19:19

Notation - meaning of mod(x,y)

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