Higher-order residue ?

[Max A.]

I believe these definitions deals only with orders dividing p-1.
In particular, "Primes p such that 2 is a cubic residue modulo p, but
not a higher-order residue" essentially means such p that
2^((p-1)/3)==1 (mod p) but for any larger divisor k of p-1,
2^((p-1)/k)!=1 (mod p).

I think that's bad formulation. Much simpler and clearer definition would be
"Primes p such that the multiplicative order of 2 modulo p is (p-1)/3".

Max

On 7/9/06, Giovanni Resta <g.resta at iit.cnr.it> wrote:
> I beg your pardon for asking a stupid question, but  (maybe it
> is the hot sun of July) there is something very basilar
> I do not get in the definition of these sequences:
>
> A001133
> Primes p such that 2 is a cubic residue modulo p, but not a higher-order residue.
> 43,109,157,...
>
> and similarly sequences A001136, A115591, A001134, A001135.
>
> I thought that
> "a number t is a k-order residue modulo p if
> there exist a number m such that m^k is congruent to t modulo p,
> i.e. m^k=t (mod p).
>
> So 2 is indeed a cubic residue mod 43, since 20^3=2 (mod 43).
> But it seems to me it is also a 5-th residue, since 8^5=2 (mod 43)
> and also a 9-th residue, since 26^9=2 (mod 43) and so on
> for orders  11,13,15,17,19,23,25,27,29,31,33,37,39,41.
> So, where I'm wrong in the interpretation of the definition
> of the sequences above ?
> Thanks,
> giovanni.
>
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