Irreducible Polynomials Which Divide Trinomials Over GF (2)

The simplest linear shift registers to generate binary sequences involve only two taps, which corresponds to a trinomial over GF(2). It is therefore of interest to know which irreducible polynomials f(x) divide trinomials over GF(2), since the output sequences corresponding to f(x) can be obtained from a two-tap linear feedback shift register (with a suitable initial state) if and only if f(x) divides some trinomial t(x)=x^m+x^a+1 over GF(2). In this paper, we develop the theory of which irreducible polynomials do, or do not, divide trinomials over GF(2). Then some related problems such as Artin´s conjecture about primitive roots, and the conjectures of Blake, Gao, and Lambert, as well as of Tromp, Zhang, and Zhao are discussed