RE: [PrimeNumbers] Cubic, quartic and higher order residues
> (1) If 2 and 5 are both cubic residues of a prime p, is 10 aIf A is a cubic residue, that means there is an a such that a^3 = A.
> cubic residue please?
So yes, if A and B are cubic residues (say the cubes of a and b),
so is their product (it is the cube of ab).
> (2) Does the same apply to quatic and highr order residues?Indeed.
> (3) If p = 2 mod 3, are ALL numbers cubic residues (thisThe order of the elements divide the order of the group.
> seems to be the case for low values of p)?
The group of integers prime to p, modulo p, has order p-1.
So if p is 2 mod three, this order is not divisible by 3
and hence all values are cubic residues.
> (4) What if neither 2 nor 5 are cubic residues - what then isIt might, or might not be. 2,5 and 10 are not cubic residues mod 7
> the status of 10? It seems to me (based only on low values of
> p) that this means 10 IS a cubic residue, providing it is not
> already a primitive root.
> (5) Do similar statements to (4) apply to quadric and higherYes, the statement that it might or might not be holds; the higher
> order residues?
the degree, the less likely it is a residue.
> (6) Where can I find out more please?An elementary number theory textbook.