[GAP Forum]re: A problem simplified

[John Dixon]

Bulutoglu Dursun A Civ AFIT/ENC wrote:
> 	Dear GAP Forum;
> 	I reduced my problem to the following simpler problem:
> 	Let Z_p={0,1,2,...p-1} be the finite field with p elements and x
> be a primitive element.
> 	Also let A={1,2,3,...,(p-1)/2} and B={1,x,x^2,...,x^(p-3)/2} be
> subsets of Z_p^* then A can not be equal to B for p>7.
> 	I was wondering if you have any ideas how this could be solved.

Summing the elements in A gives (p^2-1)/4 = -1/4 and summing the 
elements in B gives (x^((p-1)/2)-1)/(x-1) = -2/(x-1) since x is a 
quadratic nonresidue. Thus A = B would imply x = 9 which is 
impossible since a primitive root (for p > 2) cannot be a square.

John D. Dixon