I need an existence proof --- it's the missing link in a theorem establishing full-plane integer zero-free number walls over finite characteristic ( |F_2 excepted). CONJECTURE: Given any r > 1 , there exists over |F_2 a trinomial f(X) = X^s + X^t + 1 = g(X) h(X) , where g(X) is irreducible of degree r . This topic turns out to have been studied previously, since it is of interest in elliptic curve public-key cryptography. Such "redundant trinomials" are used to do arithmetic in binary finite fields |F^(2^r) lacking a trinomial generating polynomial: the idea is to calculate instead in a larger ring using trinomial f(X) with factor some field generator g(X) , afterwards reducing the result to the field. Of course, engineers don't care greatly about proving that such a trinomial exists for every field degree: they're just content that some have been found for every degree up to 10,000, comfortably exceeding anything currently required. I haven't uncovered a great deal about this in the literature: Christophe Doche (2005) <doche@ics.mq.edu.au> "Redundant Trinomials for Finite Fields of Characteristic 2 " http://web.science.mq.edu.au/~doche/redundant.pdf is cited by most of the rest; Ryul Kim, Wolfram Koepf <koepf@mathematik.uni-kassel.de> "Divisibility of Trinomials by Irreducible Polynomials over |F_2 " https://arxiv.org/pdf/1311.1366.pdf includes results about cofactor h(X) , along with baffling typos. If required I can post lists of sample trinomials and cofactors, currently for prime r < 4096 (in 3 hrs), also all r < 1024 (in 12 mins). Fred Lunnon