# Thread: Help understanding what this theorem means

1. ## Help understanding what this theorem means

Theorem. If $p$ is a prime and $(a,p)=1$, then the congruence $x^n$ $\equiv$ $a (mod p)$ has $(n, p-1)$ solutions or no solution according as
$a^{(p-1)/(n,p-1)} \equiv 1 (mod p)$
or not.

Can someone explain what this means linguistically? I don't understand what "according as [...] or not" means.

2. Originally Posted by paupsers
Theorem. If $p$ is a prime and $(a,p)=1$, then the congruence $x^n$ $\equiv$ $a (mod p)$ has $(n, p-1)$ solutions or no solution according as
$a^{(p-1)/(n,p-1)} \equiv 1 (mod p)$
or not.

Can someone explain what this means linguistically? I don't understand what "according as [...] or not" means.

"An integer a coprime with a prime p is an n-th root modulo p iff a raised to the power that

appears there equals 1 mod p"

Or in the language of groups: an element $a\in \left(\mathbb{Z}/p\mathbb{Z}\right)^*$ has n-th root there iff

its order divides $\frac{p-1}{(n,p-1)}$

Tonio

3. Originally Posted by paupsers
Theorem. If $p$ is a prime and $(a,p)=1$, then the congruence $x^n$ $\equiv$ $a (mod p)$ has $(n, p-1)$ solutions or no solution according as
$a^{(p-1)/(n,p-1)} \equiv 1 (mod p)$
or not.

Can someone explain what this means linguistically? I don't understand what "according as [...] or not" means.
this means that if $a^{(p-1)/(n,p-1)} \equiv 1 \, (mod \, p)$ then $x^n \equiv a \,(mod \, p)$ has $gcd(n,p-1)$ solutions and if $a^{(p-1)/(n,p-1)} \neq 1 \, (mod \, p)$ then $x^n \equiv a \, (mod \, p)$ has no solutions.