Primitive polynomials with first and second coefficients prescribed

Let Fq denote the finite field of q elements, q an odd prime power, and let f(x)=xn+∑i=1n fixn−i Fq[x] be given. When n 7, Han has stated that if f1 and f2 are prescribed in advance, there exist coefficients f3,…,fn Fq such that f(x) is primitive. The remaining cases of interest are n=4, 5, and 6. In this note we make some needed modifications to Hanʹs arguments for n 7, resolve the cases n=5 and 6 in the affirmative, and discuss the case n=4 as well. Our results are obtained by means of the bounds given in Hanʹs paper as well as use of sieving techniques due to the first author.