[GAP Forum] Quadratic residue in finite field

[Surinder Kaur]

If F is a field with q = p^m -1 elements where m is even and p is an odd
prime, A the set of all quadratic residues and B = {4 b - 1 | b is
primitive element of F} . Then my aim is to prove A intersection B is non
empty i.e. for some primitive element b the element 4b-1 is a square of
some element. How  to check whether the conjecture holds atleast for some p
and m's.