Let h(Y)=Y(Y-1). For every integer y, h(y) is even because one of two consecutive integers is even. Repeated polynomial division by the monic quadratic h writes p_ell(Y)=sum_{j=0}^{ell-1} e_j(Y)h(Y)^j with integer e_j of degree at most one. Monic means the leading coefficient is one, so division introduces no fractions. Degree(p_ell)<2ell ensures at most ell remainders. For ell=2, p_2(Y)=1-3Y^2+2Y^3=(1-Y)+(2Y-1)h(Y). The jth term therefore contains at least j factors of two at integer inputs. This will allow a single degree budget across many equations.