LAUSR

We define Euclid polynomials $$E_{k+1}(\lambda ) = E_{k}(\lambda )\left( E_{k}(\lambda ) - 1\right) + 1$$Ek+1(λ)=Ek(λ)Ek(λ)-1+1 and $$E_{1}(\lambda ) = \lambda + 1$$E1(λ)=λ+1 in analogy to Euclid numbers $$e_k = E_{k}(1)$$ek=Ek(1). We show how to construct companion matrices $$\mathbb {E}_k$$Ek, so $$E_k(\lambda ) = {\text {det}}\left( \lambda \mathbf {I} - \mathbb {E}_{k}\right) $$Ek(λ)=detλI-Ek, of height 1 (and thus of minimal height over all integer companion matrices for $$E_{k}(\lambda )$$Ek(λ)). We prove various properties of these objects, and give experimental confirmation of some unproved properties.