8. Euler’s prime-producing polynomial 19
Figure 2. (Based on [Zau83].)
We can assume that N (π1) ≤ N (ρ1). (If this does not hold initially,
interchange the two factorizations.) For ξ, γ ∈ Z[η] still to be chosen, define
(1.10) α := (ρ1ξ − π1γ)ρ2 · · · ρj.
α = αξ − π1
= π1(π2 · · · πkξ − ρ2 · · · ρjγ).
Factoring the parenthetical expression, we deduce that α has a factorization
into irreducibles where one of the irreducibles is π1. We will choose ξ and γ
so that π1 ρ1ξ. Then π1 ρ1ξ−π1γ, and so we may deduce from (1.10) that
α has a factorization into irreducibles, none of which is a unit multiple of
π1. So α possesses two distinct factorizations into irreducibles. If further,
γ and ξ satisfy
N (ρ1ξ − π1γ) N (ρ1),
then N (α ) is smaller than N (α), and so we have a contradiction to our
choice of α.
So it remains to show that it is possible to choose ξ, γ ∈ Z[η] with the
following two properties:
(P1) π1 ρ1ξ,
(P2) N (ρ1ξ − π1γ) N (ρ1), or equivalently, ξ −
Since N (π1) ≤ N (ρ1), the complex number π1/ρ1 lies on or inside the unit
circle. Suppose first that π1/ρ1 lies outside the shaded region indicated in