Re: Primality with respect to multiplication modulo n.
- --- In firstname.lastname@example.org,
"Dimiter Skordev" <skordev@...> wrote:
> I decided to submit the statement as a problem to the AmericanNo problem, Dimeter.
> Mathematical Monthly. I actually submitted there a little more
> precise form of that statement, but yesterday I withdraw my
> submission, since I observed that already on January 11
> David Broadhurst has posted a proof of the statement from my
> message (I am sorry for observing David's message so late!).
> I am now free to post my proofThanks. Where you write
> Suppose n is some of the numbers 2, 3, 4, 6. It easy to checkI had written:
> that, whenever x and y are natural numbers less than n and
> different from 1, the number x*y is not congruent to n-1 modulo n.
> For n = 3, 4, or 6, the group of units (Z/Zn)* haswhich avoids having to use your detailed checks, value by value.
> precisely 2 elements
That gave the key to my proof of the main theorem:
> Theorem 3: Lemmas 1 and 2 exhaust the situations in which aand then we have only to write tidy words, as you did.
> prime p < n is a "prime mod n", as defined by Skordev.
> Proof: First, suppose that n is coprime to p. If n = 5 or
> n > 6, then the group of units (Z/Zn)* has at least 4 elements