The usual definition of a primitive root modulo N is a generator of
(Z/NZ)^*, which exists iff that group is cyclic, which is iff N is an
odd prime power or double such, or 2 or 4. The existing behaviour is
consistent with that.
However for N = 2^e for e>=3 the croup is "half cyclic", i.e. there is
a number which generates a cyclic subgroup of half the order of the
group (i.e. has order 2^{e-2} while the group has order 2^{e-1}). And
5 has this property for all e>=3.
Perhaps it is that property of 5 modulo powers of 2 which you were thinking of?
The lecture ends...here!
John
On Sat, Nov 27, 2010 at 2:13 AM, Donald Alan Morrison
<[email protected]> wrote:
> On Nov 26, 5:44 pm, Donald Alan Morrison <[email protected]>
> wrote:
>> primitive_root(ZZ) ArithmeticError for (+-)2^i for i >= 3
>>
>> http://trac.sagemath.org/sage_trac/ticket/10343
>
> Please disregard. I guess there must be a more modern meaning for
> primitive roots (only the odd primes considered for p^i and 2*p^i with
> the exceptions of n=2 or n=4?)
>
> sage: for a in range(16):
> ....: print mod(5^a, 16)
> ....:
> 1
> 5
> 9
> 13
> 1
> 5
> 9
> 13
> 1
> 5
> 9
> 13
> 1
> 5
> 9
> 13
> sage:
>
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