Strongly primitive elements

Let n be a positive integer. A nonzero element gamma of the finite field F of order q = 2(n) is said to be "strongly primitive" if every element (a gamma+b)1(c gamma +d), with a, b, c, d in {0, 1} and ad - bc not zero, is primitive in the usual sense. We show that the number N of such strongly primitive elements is asymptotic to theta theta' center dot q where theta is the product of (1 - 2/p) over all primes p dividing (q - 1) and theta' is the product of (1 - 2/p) over the same set. Using this result and the accompanying error estimates, with some computer assistance for small n, we deduce the existence of such strongly primitive elements for all n except n = 1, 4, 6. This extends earlier work on Golomb's conjecture concerning the simultaneous primitivity of gamma and gamma + 1. We also discuss analogous questions concerning strong primitivity for other finite fields.