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From: recoder on 6 Apr 2010 09:27
I need to find what x would be in 3^x = 1 mod (p) where p is a prime?
Can we express x in terms of p?

Thanks in advance...
From: Dan on 6 Apr 2010 09:52
On Apr 6, 9:27 am, recoder <kurtulmeh...(a)gmail.com> wrote:
> I need to find  what x would be in 3^x = 1 mod (p) where p is a prime?
> Can we express x in terms of p?
>
> Thanks in advance...

x = 0 (mod p-1) gives a subset of the set of all solutions, by
Fermat's Little Theorem.

There's a conjecture that x = 0 (mod p-1) gives the full set of
solutions for infinitely many (but nevertheless, not that many) p:
http://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots
From: Chip Eastham on 6 Apr 2010 10:11
On Apr 6, 9:52 am, Dan <dbri...(a)mit.edu> wrote:
> On Apr 6, 9:27 am, recoder <kurtulmeh...(a)gmail.com> wrote:
>
> > I need to find  what x would be in 3^x = 1 mod (p) where p is a prime?
> > Can we express x in terms of p?
>
> > Thanks in advance...
>
> x = 0 (mod p-1) gives a subset of the set of all solutions, by
> Fermat's Little Theorem.
>
> There's a conjecture that x = 0 (mod p-1) gives the full set of
> solutions for infinitely many (but nevertheless, not that many) p:
>http://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots

Consider for example 3^x = 1 mod 7.
Per Fermat's Little Theorem,
3^6 = 1 mod 7, and in fact x = 6 is
the smallest positive integer solution.

However 3^7 = 1 mod 1093, showing that
as Dan implies above, the smallest
exponent can be a proper divisor of p-1.

regards, chip
From: recoder on 6 Apr 2010 10:58
On 6 Nisan, 17:11, Chip Eastham <hardm...(a)gmail.com> wrote:
> On Apr 6, 9:52 am, Dan <dbri...(a)mit.edu> wrote:
>
> > On Apr 6, 9:27 am, recoder <kurtulmeh...(a)gmail.com> wrote:
>
> > > I need to find  what x would be in 3^x = 1 mod (p) where p is a prime?
> > > Can we express x in terms of p?
>
> > > Thanks in advance...
>
> > x = 0 (mod p-1) gives a subset of the set of all solutions, by
> > Fermat's Little Theorem.
>
> > There's a conjecture that x = 0 (mod p-1) gives the full set of
> > solutions for infinitely many (but nevertheless, not that many) p:
> >http://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots
>
> Consider for example 3^x = 1 mod 7.
> Per Fermat's Little Theorem,
> 3^6 = 1 mod 7, and in fact x = 6 is
> the smallest positive integer solution.
>
> However 3^7 = 1 mod 1093, showing that
> as Dan implies above, the smallest
> exponent can be a proper divisor of p-1.
>
> regards, chip

Thanks a lot. I wonder if there is a solution for 3^x = y mod (p)
where y= 2^z and z is integer.
From: Pubkeybreaker on 6 Apr 2010 11:19
On Apr 6, 9:27 am, recoder <kurtulmeh...(a)gmail.com> wrote:
> I need to find  what x would be in 3^x = 1 mod (p) where p is a prime?
> Can we express x in terms of p?
>
> Thanks in advance...

All you can is that x will be a divisor of p-1 (via Lagrange's Thm)
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