Problem understanding the proof of Pocklington's criterion

Asked Jul 19 '23 at 11:29

Active Jul 19 '23 at 11:58

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I have a problem in the proof of the Pocklington criteria. The theorem itself is

Now I have this proof (my question concerns a point in the last three lines):

My question is: So we choose $p$ to be a prime divisor $n$. So $\varphi(p) = p-1$. I know that $a^{n-1}\equiv 1\pmod p$. Now the proof says that by euler's totient theorem we can deduce $r\mid p-1$. But I do not understand why. Shouldn't we assume that $p$ and $a$ coprime to deduce this?

edited Jul 19 '23 at 11:58

asked Jul 19 '23 at 11:29

mathquester

1

$a^{n-1}\equiv 1\pmod{p}\Rightarrow p\nmid a,,$ so $, a^{p-1}\equiv 1\pmod{p}\Rightarrow r:= {\rm ord}_p(a)\mid p-1,$ by Corollary$_1$ in linked dupe. – Bill Dubuque Jul 19 '23 at 14:15

Problem understanding the proof of Pocklington's criterion

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