Re: group theory question

Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:

On 09/10/2024 17:21, Phil Carmody wrote:

Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:

Well if the conjecture fails, a counter-example suffices.  But like I
said, I'm not sure what you're asking.  It should be apparent from
tests that some (p,g) values work and some do not.

>
Mod(3,17) is a generator, but its 2^n-th powers hit Mod(1,17) really
quickly.

>
Also p=13.  Well, the powers don't hit 1, but quickly cycle with a
period of 2.  Cycling "too soon" is the general way different primes
fail, rather than specifically hitting 1.

Yeah, I chose 17 because it is a Fermat prime (of the form 2^n+1),
and I knew 2^n modulo EulerPhi(17)=16 would go 1, 2, 4, 8, 0, ...

Just by inspection (mucking about in an Excel spreadsheet) it seems
most p won't work due to 2^n [mod p-1] quickly hitting a cycle, but a
few p DO work, so there's an interesting question as to why.
>
Working p:  (2), 3, 5, 7, 11, 23, 59, ...
>
(Then again I might have mucked up the spreadsheet or just misrecorded
something, so don't take that as gospel!)

OEIS is the place to look for such sequences. It's probably something
trivial.

Phil
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